{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Classifying handwritten digits using the SVD"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Load data from the MNIST data set"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {},
   "outputs": [],
   "source": [
    "using Flux\n",
    "using Flux.Data.MNIST"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "metadata": {},
   "outputs": [],
   "source": [
    "labels = MNIST.labels();\n",
    "images = MNIST.images(); "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "\u001b[32m\u001b[1m Resolving\u001b[22m\u001b[39m package versions...\n",
      "\u001b[32m\u001b[1m  Updating\u001b[22m\u001b[39m `~/.julia/environments/v1.0/Project.toml`\n",
      "\u001b[90m [no changes]\u001b[39m\n",
      "\u001b[32m\u001b[1m  Updating\u001b[22m\u001b[39m `~/.julia/environments/v1.0/Manifest.toml`\n",
      "\u001b[90m [no changes]\u001b[39m\n"
     ]
    }
   ],
   "source": [
    "using Pkg\n",
    "Pkg.add(\"ImageShow\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "metadata": {},
   "outputs": [],
   "source": [
    "using ImageShow"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 100,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(60000,)"
      ]
     },
     "execution_count": 100,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "size(images)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 102,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(28, 28)"
      ]
     },
     "execution_count": 102,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "size(images[8])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 103,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "784"
      ]
     },
     "execution_count": 103,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "28*28"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's see what the images look like:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<table><tbody><tr><td style='text-align:center;vertical-align:middle; margin: 0.5em;border:1px #90999f solid;border-collapse:collapse'><img style='max-width: 100px; max-height:100px;display:inline' src=\"data:image/png;base64,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\"></td><td style='text-align:center;vertical-align:middle; margin: 0.5em;border:1px #90999f solid;border-collapse:collapse'><img style='max-width: 100px; max-height:100px;display:inline' src=\"data:image/png;base64,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\"></td><td style='text-align:center;vertical-align:middle; margin: 0.5em;border:1px #90999f solid;border-collapse:collapse'><img style='max-width: 100px; max-height:100px;display:inline' src=\"data:image/png;base64,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\"></td><td style='text-align:center;vertical-align:middle; margin: 0.5em;border:1px #90999f solid;border-collapse:collapse'><img style='max-width: 100px; max-height:100px;display:inline' src=\"data:image/png;base64,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\"></td><td style='text-align:center;vertical-align:middle; margin: 0.5em;border:1px #90999f solid;border-collapse:collapse'><img style='max-width: 100px; max-height:100px;display:inline' src=\"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAHAAAABwCAAAAADji6uXAAAABGdBTUEAALGPC/xhBQAAACBjSFJNAAB6JgAAgIQAAPoAAACA6AAAdTAAAOpgAAA6mAAAF3CculE8AAAAAmJLR0QA/4ePzL8AAAHwSURBVGje7dnPi41RHMfx10j5kSShbCwMG2QjWRkbsZNYjAWFnYWdkpKN8QfY2LAgqWlEyWxk49dmyk4ZJVaajZKUSJlr8T23mabmzr2Pmnvu6Xw29/l1nvf93M/9PufHQ1VVVVVVVVVVVVVV1f9rqNcGB3AGI9idjl3CDA7iPqY6tF+x3A7LB/aU4ShuYlNq+AKbsWvezR7iVE4Oyweu7Pai/biNtXiF63iDVZjAkXTt29wclg/sKsPTuJO2n4t6/JH2R83l9wX3cnNYPnDJZ+kYrqCFW7hqLj+Yxs60fRJPcnNYPrBjHV4T+f3BM1zGr3Rutai/beKPMGbp/PrisHzgonW4AR/E+GUSx+ed24EH2Jf2H+E8fubosHzgohluEfMF2I7fOIdj2IN14vnawgk8zdVh+cCOdTgt5g5DIqu2ZtKxrfiaPrN1WD5w0f7wu3h+TmIjPon+7i6+YVxkN567w/KBHcc0U6IOF2oEhzCLz7k7LB/Y1fxwodaI/FpqHWYA7Hm9tK2/IsN2n5itw/KBjerw6CA5LB/YKMPhQXJYPrBRhq/TN50dBIflAxtl+A4fxdx/WO0P+wxsPKY5K95lvMRFvM/VYfnAxhmuF+8ND+OxWIer66V9ATbOkMjxBi5gr+5qsfyfdNmBVYOvfxOISKianOH0AAAAAElFTkSuQmCC\"></td></tr></tbody></table><div><small>(a vector displayed as a row to save space)</small></div>"
      ],
      "text/plain": [
       "5-element Array{Array{ColorTypes.Gray{FixedPointNumbers.Normed{UInt8,8}},2},1}:\n",
       " [Gray{N0f8}(0.0) Gray{N0f8}(0.0) … Gray{N0f8}(0.0) Gray{N0f8}(0.0); Gray{N0f8}(0.0) Gray{N0f8}(0.0) … Gray{N0f8}(0.0) Gray{N0f8}(0.0); … ; Gray{N0f8}(0.0) Gray{N0f8}(0.0) … Gray{N0f8}(0.0) Gray{N0f8}(0.0); Gray{N0f8}(0.0) Gray{N0f8}(0.0) … Gray{N0f8}(0.0) Gray{N0f8}(0.0)]\n",
       " [Gray{N0f8}(0.0) Gray{N0f8}(0.0) … Gray{N0f8}(0.0) Gray{N0f8}(0.0); Gray{N0f8}(0.0) Gray{N0f8}(0.0) … Gray{N0f8}(0.0) Gray{N0f8}(0.0); … ; Gray{N0f8}(0.0) Gray{N0f8}(0.0) … Gray{N0f8}(0.0) Gray{N0f8}(0.0); Gray{N0f8}(0.0) Gray{N0f8}(0.0) … Gray{N0f8}(0.0) Gray{N0f8}(0.0)]\n",
       " [Gray{N0f8}(0.0) Gray{N0f8}(0.0) … Gray{N0f8}(0.0) Gray{N0f8}(0.0); Gray{N0f8}(0.0) Gray{N0f8}(0.0) … Gray{N0f8}(0.0) Gray{N0f8}(0.0); … ; Gray{N0f8}(0.0) Gray{N0f8}(0.0) … Gray{N0f8}(0.0) Gray{N0f8}(0.0); Gray{N0f8}(0.0) Gray{N0f8}(0.0) … Gray{N0f8}(0.0) Gray{N0f8}(0.0)]\n",
       " [Gray{N0f8}(0.0) Gray{N0f8}(0.0) … Gray{N0f8}(0.0) Gray{N0f8}(0.0); Gray{N0f8}(0.0) Gray{N0f8}(0.0) … Gray{N0f8}(0.0) Gray{N0f8}(0.0); … ; Gray{N0f8}(0.0) Gray{N0f8}(0.0) … Gray{N0f8}(0.0) Gray{N0f8}(0.0); Gray{N0f8}(0.0) Gray{N0f8}(0.0) … Gray{N0f8}(0.0) Gray{N0f8}(0.0)]\n",
       " [Gray{N0f8}(0.0) Gray{N0f8}(0.0) … Gray{N0f8}(0.0) Gray{N0f8}(0.0); Gray{N0f8}(0.0) Gray{N0f8}(0.0) … Gray{N0f8}(0.0) Gray{N0f8}(0.0); … ; Gray{N0f8}(0.0) Gray{N0f8}(0.0) … Gray{N0f8}(0.0) Gray{N0f8}(0.0); Gray{N0f8}(0.0) Gray{N0f8}(0.0) … Gray{N0f8}(0.0) Gray{N0f8}(0.0)]"
      ]
     },
     "execution_count": 37,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "images[1:5]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 104,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<table><tbody><tr><td style='text-align:center;vertical-align:middle; margin: 0.5em;border:1px #90999f solid;border-collapse:collapse'><img style='max-width: 100px; max-height:100px;display:inline' src=\"data:image/png;base64,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\"></td><td style='text-align:center;vertical-align:middle; margin: 0.5em;border:1px #90999f solid;border-collapse:collapse'><img style='max-width: 100px; max-height:100px;display:inline' src=\"data:image/png;base64,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\"></td><td style='text-align:center;vertical-align:middle; margin: 0.5em;border:1px #90999f solid;border-collapse:collapse'><img style='max-width: 100px; max-height:100px;display:inline' src=\"data:image/png;base64,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\"></td><td style='text-align:center;vertical-align:middle; 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margin: 0.5em;border:1px #90999f solid;border-collapse:collapse'><img style='max-width: 100px; max-height:100px;display:inline' src=\"data:image/png;base64,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\"></td><td style='text-align:center;vertical-align:middle; margin: 0.5em;border:1px #90999f solid;border-collapse:collapse'><img style='max-width: 100px; max-height:100px;display:inline' src=\"data:image/png;base64,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\"></td><td style='text-align:center;vertical-align:middle; margin: 0.5em;border:1px #90999f solid;border-collapse:collapse'><img style='max-width: 100px; max-height:100px;display:inline' src=\"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAHAAAABwCAAAAADji6uXAAAABGdBTUEAALGPC/xhBQAAACBjSFJNAAB6JgAAgIQAAPoAAACA6AAAdTAAAOpgAAA6mAAAF3CculE8AAAAAmJLR0QA/4ePzL8AAAGMSURBVGje7dmhS11hGMfxj2JyWFamQVgQBG+XFatNMM1gGxZZWdPFIRYHkwUNE/8DDTNYRLQtCgqOtVkGYltyyNBwDniT91zPRd/z8P7S88LlfM+XH8+5h3vJycnJycnJyWl++np1oY9YxRqWH/hc/1Mbxgf2pMMh/MIr3OA9tlMxjA8c6MUFFhX9wSV+pGQYH1h7D6dw3HaewX5KhvGBtfbwNb62nQ9xlJphfGCtPTzDRDn/xVscpGYYH1hrD1u4LedNnft7FsP4wEd3+EWxxLeKZ+hKqobxgY/qcAOziv5OMY/rVA3jA7vucFLR33B5/oarlA3jA7vu8B1GyvknvqduGB/Y1XvpB3xuu8tR/EndMD6w8h6OYqG8w//Y0n1/z2IYH1ipwzHsYbw8r2OpKYbxgZU6HHffH0WfjTGMD6zU4cu2+RjnTTKMD6z0TvNb8X0Ic9hpkmF8YMc9bOFFOX/CbtMM4wM7dvhG8R8v/HP/+2hjDOMDKz1LLzCIaZw0zTA+MCcnJyenfu4APw4t2kRtTgAAAAAASUVORK5C\"></td></tr><tr><td style='text-align:center;vertical-align:middle; margin: 0.5em;border:1px #90999f solid;border-collapse:collapse'><img style='max-width: 100px; max-height:100px;display:inline' src=\"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAHAAAABwCAAAAADji6uXAAAABGdBTUEAALGPC/xhBQAAACBjSFJNAAB6JgAAgIQAAPoAAACA6AAAdTAAAOpgAAA6mAAAF3CculE8AAAAAmJLR0QA/4ePzL8AAAHwSURBVGje7dnPi41RHMfx10j5kSShbCwMG2QjWRkbsZNYjAWFnYWdkpKN8QfY2LAgqWlEyWxk49dmyk4ZJVaajZKUSJlr8T23mabmzr2Pmnvu6Xw29/l1nvf93M/9PufHQ1VVVVVVVVVVVVVV1f9rqNcGB3AGI9idjl3CDA7iPqY6tF+x3A7LB/aU4ShuYlNq+AKbsWvezR7iVE4Oyweu7Pai/biNtXiF63iDVZjAkXTt29wclg/sKsPTuJO2n4t6/JH2R83l9wX3cnNYPnDJZ+kYrqCFW7hqLj+Yxs60fRJPcnNYPrBjHV4T+f3BM1zGr3Rutai/beKPMGbp/PrisHzgonW4AR/E+GUSx+ed24EH2Jf2H+E8fubosHzgohluEfMF2I7fOIdj2IN14vnawgk8zdVh+cCOdTgt5g5DIqu2ZtKxrfiaPrN1WD5w0f7wu3h+TmIjPon+7i6+YVxkN567w/KBHcc0U6IOF2oEhzCLz7k7LB/Y1fxwodaI/FpqHWYA7Hm9tK2/IsN2n5itw/KBjerw6CA5LB/YKMPhQXJYPrBRhq/TN50dBIflAxtl+A4fxdx/WO0P+wxsPKY5K95lvMRFvM/VYfnAxhmuF+8ND+OxWIer66V9ATbOkMjxBi5gr+5qsfyfdNmBVYOvfxOISKianOH0AAAAAElFTkSuQmCC\"></td><td style='text-align:center;vertical-align:middle; margin: 0.5em;border:1px #90999f solid;border-collapse:collapse'><img style='max-width: 100px; max-height:100px;display:inline' src=\"data:image/png;base64,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\"></td><td style='text-align:center;vertical-align:middle; margin: 0.5em;border:1px #90999f solid;border-collapse:collapse'><img style='max-width: 100px; max-height:100px;display:inline' src=\"data:image/png;base64,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\"></td><td style='text-align:center;vertical-align:middle; margin: 0.5em;border:1px #90999f solid;border-collapse:collapse'><img style='max-width: 100px; max-height:100px;display:inline' src=\"data:image/png;base64,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\"></td><td style='text-align:center;vertical-align:middle; margin: 0.5em;border:1px #90999f solid;border-collapse:collapse'><img style='max-width: 100px; max-height:100px;display:inline' src=\"data:image/png;base64,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\"></td></tr></tbody></table>"
      ],
      "text/plain": [
       "5×5 Array{Array{Gray{FixedPointNumbers.Normed{UInt8,8}},2},2}:\n",
       " [Gray{N0f8}(0.0) Gray{N0f8}(0.0) … Gray{N0f8}(0.0) Gray{N0f8}(0.0); Gray{N0f8}(0.0) Gray{N0f8}(0.0) … Gray{N0f8}(0.0) Gray{N0f8}(0.0); … ; Gray{N0f8}(0.0) Gray{N0f8}(0.0) … Gray{N0f8}(0.0) Gray{N0f8}(0.0); Gray{N0f8}(0.0) Gray{N0f8}(0.0) … Gray{N0f8}(0.0) Gray{N0f8}(0.0)]  …  [Gray{N0f8}(0.0) Gray{N0f8}(0.0) … Gray{N0f8}(0.0) Gray{N0f8}(0.0); Gray{N0f8}(0.0) Gray{N0f8}(0.0) … Gray{N0f8}(0.0) Gray{N0f8}(0.0); … ; Gray{N0f8}(0.0) Gray{N0f8}(0.0) … Gray{N0f8}(0.0) Gray{N0f8}(0.0); Gray{N0f8}(0.0) Gray{N0f8}(0.0) … Gray{N0f8}(0.0) Gray{N0f8}(0.0)]\n",
       " [Gray{N0f8}(0.0) Gray{N0f8}(0.0) … Gray{N0f8}(0.0) Gray{N0f8}(0.0); Gray{N0f8}(0.0) Gray{N0f8}(0.0) … Gray{N0f8}(0.0) Gray{N0f8}(0.0); … ; Gray{N0f8}(0.0) Gray{N0f8}(0.0) … Gray{N0f8}(0.0) Gray{N0f8}(0.0); Gray{N0f8}(0.0) Gray{N0f8}(0.0) … Gray{N0f8}(0.0) Gray{N0f8}(0.0)]     [Gray{N0f8}(0.0) Gray{N0f8}(0.0) … Gray{N0f8}(0.0) Gray{N0f8}(0.0); Gray{N0f8}(0.0) Gray{N0f8}(0.0) … Gray{N0f8}(0.0) Gray{N0f8}(0.0); … ; Gray{N0f8}(0.0) Gray{N0f8}(0.0) … Gray{N0f8}(0.0) Gray{N0f8}(0.0); Gray{N0f8}(0.0) Gray{N0f8}(0.0) … Gray{N0f8}(0.0) Gray{N0f8}(0.0)]\n",
       " [Gray{N0f8}(0.0) Gray{N0f8}(0.0) … Gray{N0f8}(0.0) Gray{N0f8}(0.0); Gray{N0f8}(0.0) Gray{N0f8}(0.0) … Gray{N0f8}(0.0) Gray{N0f8}(0.0); … ; Gray{N0f8}(0.0) Gray{N0f8}(0.0) … Gray{N0f8}(0.0) Gray{N0f8}(0.0); Gray{N0f8}(0.0) Gray{N0f8}(0.0) … Gray{N0f8}(0.0) Gray{N0f8}(0.0)]     [Gray{N0f8}(0.0) Gray{N0f8}(0.0) … Gray{N0f8}(0.0) Gray{N0f8}(0.0); Gray{N0f8}(0.0) Gray{N0f8}(0.0) … Gray{N0f8}(0.0) Gray{N0f8}(0.0); … ; Gray{N0f8}(0.0) Gray{N0f8}(0.0) … Gray{N0f8}(0.0) Gray{N0f8}(0.0); Gray{N0f8}(0.0) Gray{N0f8}(0.0) … Gray{N0f8}(0.0) Gray{N0f8}(0.0)]\n",
       " [Gray{N0f8}(0.0) Gray{N0f8}(0.0) … Gray{N0f8}(0.0) Gray{N0f8}(0.0); Gray{N0f8}(0.0) Gray{N0f8}(0.0) … Gray{N0f8}(0.0) Gray{N0f8}(0.0); … ; Gray{N0f8}(0.0) Gray{N0f8}(0.0) … Gray{N0f8}(0.0) Gray{N0f8}(0.0); Gray{N0f8}(0.0) Gray{N0f8}(0.0) … Gray{N0f8}(0.0) Gray{N0f8}(0.0)]     [Gray{N0f8}(0.0) Gray{N0f8}(0.0) … Gray{N0f8}(0.0) Gray{N0f8}(0.0); Gray{N0f8}(0.0) Gray{N0f8}(0.0) … Gray{N0f8}(0.0) Gray{N0f8}(0.0); … ; Gray{N0f8}(0.0) Gray{N0f8}(0.0) … Gray{N0f8}(0.0) Gray{N0f8}(0.0); Gray{N0f8}(0.0) Gray{N0f8}(0.0) … Gray{N0f8}(0.0) Gray{N0f8}(0.0)]\n",
       " [Gray{N0f8}(0.0) Gray{N0f8}(0.0) … Gray{N0f8}(0.0) Gray{N0f8}(0.0); Gray{N0f8}(0.0) Gray{N0f8}(0.0) … Gray{N0f8}(0.0) Gray{N0f8}(0.0); … ; Gray{N0f8}(0.0) Gray{N0f8}(0.0) … Gray{N0f8}(0.0) Gray{N0f8}(0.0); Gray{N0f8}(0.0) Gray{N0f8}(0.0) … Gray{N0f8}(0.0) Gray{N0f8}(0.0)]     [Gray{N0f8}(0.0) Gray{N0f8}(0.0) … Gray{N0f8}(0.0) Gray{N0f8}(0.0); Gray{N0f8}(0.0) Gray{N0f8}(0.0) … Gray{N0f8}(0.0) Gray{N0f8}(0.0); … ; Gray{N0f8}(0.0) Gray{N0f8}(0.0) … Gray{N0f8}(0.0) Gray{N0f8}(0.0); Gray{N0f8}(0.0) Gray{N0f8}(0.0) … Gray{N0f8}(0.0) Gray{N0f8}(0.0)]"
      ]
     },
     "execution_count": 104,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "reshape(images[1:25], 5, 5)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 39,
   "metadata": {},
   "outputs": [],
   "source": [
    "using Interact"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 107,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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",
      "text/plain": [
       "28×28 Array{Gray{N0f8},2} with eltype Gray{FixedPointNumbers.Normed{UInt8,8}}:\n",
       " Gray{N0f8}(0.0)  Gray{N0f8}(0.0)  …  Gray{N0f8}(0.0)  Gray{N0f8}(0.0)\n",
       " Gray{N0f8}(0.0)  Gray{N0f8}(0.0)     Gray{N0f8}(0.0)  Gray{N0f8}(0.0)\n",
       " Gray{N0f8}(0.0)  Gray{N0f8}(0.0)     Gray{N0f8}(0.0)  Gray{N0f8}(0.0)\n",
       " Gray{N0f8}(0.0)  Gray{N0f8}(0.0)     Gray{N0f8}(0.0)  Gray{N0f8}(0.0)\n",
       " Gray{N0f8}(0.0)  Gray{N0f8}(0.0)     Gray{N0f8}(0.0)  Gray{N0f8}(0.0)\n",
       " Gray{N0f8}(0.0)  Gray{N0f8}(0.0)  …  Gray{N0f8}(0.0)  Gray{N0f8}(0.0)\n",
       " Gray{N0f8}(0.0)  Gray{N0f8}(0.0)     Gray{N0f8}(0.0)  Gray{N0f8}(0.0)\n",
       " Gray{N0f8}(0.0)  Gray{N0f8}(0.0)     Gray{N0f8}(0.0)  Gray{N0f8}(0.0)\n",
       " Gray{N0f8}(0.0)  Gray{N0f8}(0.0)     Gray{N0f8}(0.0)  Gray{N0f8}(0.0)\n",
       " Gray{N0f8}(0.0)  Gray{N0f8}(0.0)     Gray{N0f8}(0.0)  Gray{N0f8}(0.0)\n",
       " Gray{N0f8}(0.0)  Gray{N0f8}(0.0)  …  Gray{N0f8}(0.0)  Gray{N0f8}(0.0)\n",
       " Gray{N0f8}(0.0)  Gray{N0f8}(0.0)     Gray{N0f8}(0.0)  Gray{N0f8}(0.0)\n",
       " Gray{N0f8}(0.0)  Gray{N0f8}(0.0)     Gray{N0f8}(0.0)  Gray{N0f8}(0.0)\n",
       " ⋮                                 ⋱                                  \n",
       " Gray{N0f8}(0.0)  Gray{N0f8}(0.0)     Gray{N0f8}(0.0)  Gray{N0f8}(0.0)\n",
       " Gray{N0f8}(0.0)  Gray{N0f8}(0.0)     Gray{N0f8}(0.0)  Gray{N0f8}(0.0)\n",
       " Gray{N0f8}(0.0)  Gray{N0f8}(0.0)     Gray{N0f8}(0.0)  Gray{N0f8}(0.0)\n",
       " Gray{N0f8}(0.0)  Gray{N0f8}(0.0)     Gray{N0f8}(0.0)  Gray{N0f8}(0.0)\n",
       " Gray{N0f8}(0.0)  Gray{N0f8}(0.0)  …  Gray{N0f8}(0.0)  Gray{N0f8}(0.0)\n",
       " Gray{N0f8}(0.0)  Gray{N0f8}(0.0)     Gray{N0f8}(0.0)  Gray{N0f8}(0.0)\n",
       " Gray{N0f8}(0.0)  Gray{N0f8}(0.0)     Gray{N0f8}(0.0)  Gray{N0f8}(0.0)\n",
       " Gray{N0f8}(0.0)  Gray{N0f8}(0.0)     Gray{N0f8}(0.0)  Gray{N0f8}(0.0)\n",
       " Gray{N0f8}(0.0)  Gray{N0f8}(0.0)     Gray{N0f8}(0.0)  Gray{N0f8}(0.0)\n",
       " Gray{N0f8}(0.0)  Gray{N0f8}(0.0)  …  Gray{N0f8}(0.0)  Gray{N0f8}(0.0)\n",
       " Gray{N0f8}(0.0)  Gray{N0f8}(0.0)     Gray{N0f8}(0.0)  Gray{N0f8}(0.0)\n",
       " Gray{N0f8}(0.0)  Gray{N0f8}(0.0)     Gray{N0f8}(0.0)  Gray{N0f8}(0.0)"
      ]
     },
     "execution_count": 107,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "images[17]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 108,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "2"
      ]
     },
     "execution_count": 108,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "labels[17]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 109,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<div class='tex2jax_ignore interactbulma'>\n",
       "<div class='display:none'></div><unsafe-script style='display:none'>\n",
       "WebIO.mount(this.previousSibling,{&quot;props&quot;:{},&quot;nodeType&quot;:&quot;DOM&quot;,&quot;type&quot;:&quot;node&quot;,&quot;instanceArgs&quot;:{&quot;namespace&quot;:&quot;html&quot;,&quot;tag&quot;:&quot;div&quot;},&quot;children&quot;:[{&quot;props&quot;:{&quot;className&quot;:&quot;field&quot;},&quot;nodeType&quot;:&quot;DOM&quot;,&quot;type&quot;:&quot;node&quot;,&quot;instanceArgs&quot;:{&quot;namespace&quot;:&quot;html&quot;,&quot;tag&quot;:&quot;div&quot;},&quot;children&quot;:[{&quot;props&quot;:{},&quot;nodeType&quot;:&quot;Scope&quot;,&quot;type&quot;:&quot;node&quot;,&quot;instanceArgs&quot;:{&quot;imports&quot;:{&quot;data&quot;:[{&quot;name&quot;:&quot;knockout&quot;,&quot;type&quot;:&quot;js&quot;,&quot;url&quot;:&quot;/assetserver/06c67466ebc6c52906ffd4101b7c74a47f9567bc-knockout.js&quot;},{&quot;name&quot;:&quot;knockout_punches&quot;,&quot;type&quot;:&quot;js&quot;,&quot;url&quot;:&quot;/assetserver/46b0ee2b2af2730ca29ddb46f091ac108fb8f609-knockout_punches.js&quot;},{&quot;name&quot;:null,&quot;type&quot;:&quot;js&quot;,&quot;url&quot;:&quot;/assetserver/e923a2cf7cc8bb183f572d980ae606c5f08fa0dc-all.js&quot;},{&quot;name&quot;:null,&quot;type&quot;:&quot;css&quot;,&quot;url&quot;:&quot;/assetserver/b973c333a7d1e5a6ca74f1442a2f26b5e4d0580e-style.css&quot;},{&quot;name&quot;:null,&quot;type&quot;:&quot;css&quot;,&quot;url&quot;:&quot;/assetserver/885f9c98e35e2ecc817c654a867b4570a606ec22-main.css&quot;}],&quot;type&quot;:&quot;async_block&quot;},&quot;id&quot;:&quot;knockout-component-b0053a2f-bc3b-45b5-b337-359219f0ffa7&quot;,&quot;handlers&quot;:{&quot;_promises&quot;:{&quot;importsLoaded&quot;:[function (ko, koPunches) {\n",
       "    ko.punches.enableAll();\n",
       "    ko.bindingHandlers.numericValue = {\n",
       "        init : function(element, valueAccessor, allBindings, data, context) {\n",
       "            var stringified = ko.observable(ko.unwrap(valueAccessor()));\n",
       "            stringified.subscribe(function(value) {\n",
       "                var val = parseFloat(value);\n",
       "                if (!isNaN(val)) {\n",
       "                    valueAccessor()(val);\n",
       "                }\n",
       "            })\n",
       "            valueAccessor().subscribe(function(value) {\n",
       "                var str = JSON.stringify(value);\n",
       "                if ((str == &quot;0&quot;) &amp;&amp; ([&quot;-0&quot;, &quot;-0.&quot;].indexOf(stringified()) &gt;= 0))\n",
       "                     return;\n",
       "                 if ([&quot;null&quot;, &quot;&quot;].indexOf(str) &gt;= 0)\n",
       "                     return;\n",
       "                stringified(str);\n",
       "            })\n",
       "            ko.applyBindingsToNode(element, { value: stringified, valueUpdate: allBindings.get(&#39;valueUpdate&#39;)}, context);\n",
       "        }\n",
       "    };\n",
       "    var json_data = JSON.parse(&quot;{\\&quot;changes\\&quot;:0,\\&quot;value\\&quot;:30000}&quot;);\n",
       "    var self = this;\n",
       "    function AppViewModel() {\n",
       "        for (var key in json_data) {\n",
       "            var el = json_data[key];\n",
       "            this[key] = Array.isArray(el) ? ko.observableArray(el) : ko.observable(el);\n",
       "        }\n",
       "        \n",
       "        \n",
       "        [this[&quot;changes&quot;].subscribe((function (val){!(this.valueFromJulia[&quot;changes&quot;]) ? (WebIO.setval({&quot;name&quot;:&quot;changes&quot;,&quot;scope&quot;:&quot;knockout-component-b0053a2f-bc3b-45b5-b337-359219f0ffa7&quot;,&quot;id&quot;:&quot;ob_76&quot;,&quot;type&quot;:&quot;observable&quot;},val)) : undefined; return this.valueFromJulia[&quot;changes&quot;]=false}),self),this[&quot;value&quot;].subscribe((function (val){!(this.valueFromJulia[&quot;value&quot;]) ? (WebIO.setval({&quot;name&quot;:&quot;value&quot;,&quot;scope&quot;:&quot;knockout-component-b0053a2f-bc3b-45b5-b337-359219f0ffa7&quot;,&quot;id&quot;:&quot;ob_75&quot;,&quot;type&quot;:&quot;observable&quot;},val)) : undefined; return this.valueFromJulia[&quot;value&quot;]=false}),self)]\n",
       "        \n",
       "    }\n",
       "    self.model = new AppViewModel();\n",
       "    self.valueFromJulia = {};\n",
       "    for (var key in json_data) {\n",
       "        self.valueFromJulia[key] = false;\n",
       "    }\n",
       "    ko.applyBindings(self.model, self.dom);\n",
       "}\n",
       "]},&quot;changes&quot;:[(function (val){return (val!=this.model[&quot;changes&quot;]()) ? (this.valueFromJulia[&quot;changes&quot;]=true, this.model[&quot;changes&quot;](val)) : undefined})],&quot;value&quot;:[(function (val){return (val!=this.model[&quot;value&quot;]()) ? (this.valueFromJulia[&quot;value&quot;]=true, this.model[&quot;value&quot;](val)) : undefined})]},&quot;systemjs_options&quot;:null,&quot;observables&quot;:{&quot;changes&quot;:{&quot;sync&quot;:false,&quot;id&quot;:&quot;ob_76&quot;,&quot;value&quot;:0},&quot;value&quot;:{&quot;sync&quot;:true,&quot;id&quot;:&quot;ob_75&quot;,&quot;value&quot;:30000}}},&quot;children&quot;:[{&quot;props&quot;:{&quot;attributes&quot;:{&quot;class&quot;:&quot;interact-flex-row&quot;}},&quot;nodeType&quot;:&quot;DOM&quot;,&quot;type&quot;:&quot;node&quot;,&quot;instanceArgs&quot;:{&quot;namespace&quot;:&quot;html&quot;,&quot;tag&quot;:&quot;div&quot;},&quot;children&quot;:[{&quot;props&quot;:{&quot;attributes&quot;:{&quot;class&quot;:&quot;interact-flex-row-left&quot;}},&quot;nodeType&quot;:&quot;DOM&quot;,&quot;type&quot;:&quot;node&quot;,&quot;instanceArgs&quot;:{&quot;namespace&quot;:&quot;html&quot;,&quot;tag&quot;:&quot;div&quot;},&quot;children&quot;:[{&quot;props&quot;:{&quot;className&quot;:&quot;interact &quot;,&quot;style&quot;:{&quot;padding&quot;:&quot;5px 10px 0px 10px&quot;}},&quot;nodeType&quot;:&quot;DOM&quot;,&quot;type&quot;:&quot;node&quot;,&quot;instanceArgs&quot;:{&quot;namespace&quot;:&quot;html&quot;,&quot;tag&quot;:&quot;label&quot;},&quot;children&quot;:[&quot;i&quot;]}]},{&quot;props&quot;:{&quot;attributes&quot;:{&quot;class&quot;:&quot;interact-flex-row-center&quot;}},&quot;nodeType&quot;:&quot;DOM&quot;,&quot;type&quot;:&quot;node&quot;,&quot;instanceArgs&quot;:{&quot;namespace&quot;:&quot;html&quot;,&quot;tag&quot;:&quot;div&quot;},&quot;children&quot;:[{&quot;props&quot;:{&quot;max&quot;:60000,&quot;min&quot;:1,&quot;attributes&quot;:{&quot;type&quot;:&quot;range&quot;,&quot;data-bind&quot;:&quot;numericValue: value, valueUpdate: &#39;input&#39;, event: {change : function () {this.changes(this.changes()+1)}}&quot;,&quot;orient&quot;:&quot;horizontal&quot;},&quot;step&quot;:1,&quot;className&quot;:&quot;slider slider is-fullwidth&quot;,&quot;style&quot;:{}},&quot;nodeType&quot;:&quot;DOM&quot;,&quot;type&quot;:&quot;node&quot;,&quot;instanceArgs&quot;:{&quot;namespace&quot;:&quot;html&quot;,&quot;tag&quot;:&quot;input&quot;},&quot;children&quot;:[]}]},{&quot;props&quot;:{&quot;attributes&quot;:{&quot;class&quot;:&quot;interact-flex-row-right&quot;}},&quot;nodeType&quot;:&quot;DOM&quot;,&quot;type&quot;:&quot;node&quot;,&quot;instanceArgs&quot;:{&quot;namespace&quot;:&quot;html&quot;,&quot;tag&quot;:&quot;div&quot;},&quot;children&quot;:[{&quot;props&quot;:{&quot;attributes&quot;:{&quot;data-bind&quot;:&quot;text: value&quot;}},&quot;nodeType&quot;:&quot;DOM&quot;,&quot;type&quot;:&quot;node&quot;,&quot;instanceArgs&quot;:{&quot;namespace&quot;:&quot;html&quot;,&quot;tag&quot;:&quot;p&quot;},&quot;children&quot;:[]}]}]}]}]},{&quot;props&quot;:{},&quot;nodeType&quot;:&quot;Scope&quot;,&quot;type&quot;:&quot;node&quot;,&quot;instanceArgs&quot;:{&quot;imports&quot;:{&quot;data&quot;:[],&quot;type&quot;:&quot;async_block&quot;},&quot;id&quot;:&quot;scope-d660dcc1-09f8-4571-8c98-0f967f9b7e78&quot;,&quot;handlers&quot;:{&quot;obs-output&quot;:[function (updated_htmlstr) {\n",
       "    var el = this.dom.querySelector(&quot;#out&quot;);\n",
       "    WebIO.propUtils.setInnerHtml(el, updated_htmlstr);\n",
       "}]},&quot;systemjs_options&quot;:null,&quot;observables&quot;:{&quot;obs-output&quot;:{&quot;sync&quot;:false,&quot;id&quot;:&quot;ob_79&quot;,&quot;value&quot;:&quot;&lt;div class=&#39;display:none&#39;&gt;&lt;/div&gt;&lt;unsafe-script style=&#39;display:none&#39;&gt;\\nWebIO.mount(this.previousSibling,{&amp;quot;props&amp;quot;:{&amp;quot;attributes&amp;quot;:{&amp;quot;class&amp;quot;:&amp;quot;interact-flex-row&amp;quot;}},&amp;quot;nodeType&amp;quot;:&amp;quot;DOM&amp;quot;,&amp;quot;type&amp;quot;:&amp;quot;node&amp;quot;,&amp;quot;instanceArgs&amp;quot;:{&amp;quot;namespace&amp;quot;:&amp;quot;html&amp;quot;,&amp;quot;tag&amp;quot;:&amp;quot;div&amp;quot;},&amp;quot;children&amp;quot;:[{&amp;quot;props&amp;quot;:{&amp;quot;style&amp;quot;:{&amp;quot;display&amp;quot;:&amp;quot;flex&amp;quot;,&amp;quot;flex-direction&amp;quot;:&amp;quot;row&amp;quot;}},&amp;quot;nodeType&amp;quot;:&amp;quot;DOM&amp;quot;,&amp;quot;type&amp;quot;:&amp;quot;node&amp;quot;,&amp;quot;instanceArgs&amp;quot;:{&amp;quot;namespace&amp;quot;:&amp;quot;html&amp;quot;,&amp;quot;tag&amp;quot;:&amp;quot;div&amp;quot;},&amp;quot;children&amp;quot;:[{&amp;quot;props&amp;quot;:{&amp;quot;setInnerHtml&amp;quot;:&amp;quot;&amp;lt;img src=&amp;#39;data:image/png;base64,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&amp;#39;&amp;gt;&amp;lt;/img&amp;gt;&amp;quot;},&amp;quot;nodeType&amp;quot;:&amp;quot;DOM&amp;quot;,&amp;quot;type&amp;quot;:&amp;quot;node&amp;quot;,&amp;quot;instanceArgs&amp;quot;:{&amp;quot;namespace&amp;quot;:&amp;quot;html&amp;quot;,&amp;quot;tag&amp;quot;:&amp;quot;div&amp;quot;},&amp;quot;children&amp;quot;:[]},{&amp;quot;props&amp;quot;:{&amp;quot;style&amp;quot;:{&amp;quot;height&amp;quot;:&amp;quot;0.0px&amp;quot;,&amp;quot;width&amp;quot;:&amp;quot;1.0em&amp;quot;}},&amp;quot;nodeType&amp;quot;:&amp;quot;DOM&amp;quot;,&amp;quot;type&amp;quot;:&amp;quot;node&amp;quot;,&amp;quot;instanceArgs&amp;quot;:{&amp;quot;namespace&amp;quot;:&amp;quot;html&amp;quot;,&amp;quot;tag&amp;quot;:&amp;quot;div&amp;quot;},&amp;quot;children&amp;quot;:[]},&amp;quot;label: 1&amp;quot;]}]})&lt;/unsafe-script&gt;&quot;}}},&quot;children&quot;:[{&quot;props&quot;:{&quot;id&quot;:&quot;out&quot;,&quot;setInnerHtml&quot;:&quot;&lt;div class=&#39;display:none&#39;&gt;&lt;/div&gt;&lt;unsafe-script style=&#39;display:none&#39;&gt;\\nWebIO.mount(this.previousSibling,{&amp;quot;props&amp;quot;:{&amp;quot;attributes&amp;quot;:{&amp;quot;class&amp;quot;:&amp;quot;interact-flex-row&amp;quot;}},&amp;quot;nodeType&amp;quot;:&amp;quot;DOM&amp;quot;,&amp;quot;type&amp;quot;:&amp;quot;node&amp;quot;,&amp;quot;instanceArgs&amp;quot;:{&amp;quot;namespace&amp;quot;:&amp;quot;html&amp;quot;,&amp;quot;tag&amp;quot;:&amp;quot;div&amp;quot;},&amp;quot;children&amp;quot;:[{&amp;quot;props&amp;quot;:{&amp;quot;style&amp;quot;:{&amp;quot;display&amp;quot;:&amp;quot;flex&amp;quot;,&amp;quot;flex-direction&amp;quot;:&amp;quot;row&amp;quot;}},&amp;quot;nodeType&amp;quot;:&amp;quot;DOM&amp;quot;,&amp;quot;type&amp;quot;:&amp;quot;node&amp;quot;,&amp;quot;instanceArgs&amp;quot;:{&amp;quot;namespace&amp;quot;:&amp;quot;html&amp;quot;,&amp;quot;tag&amp;quot;:&amp;quot;div&amp;quot;},&amp;quot;children&amp;quot;:[{&amp;quot;props&amp;quot;:{&amp;quot;setInnerHtml&amp;quot;:&amp;quot;&amp;lt;img src=&amp;#39;data:image/png;base64,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&amp;#39;&amp;gt;&amp;lt;/img&amp;gt;&amp;quot;},&amp;quot;nodeType&amp;quot;:&amp;quot;DOM&amp;quot;,&amp;quot;type&amp;quot;:&amp;quot;node&amp;quot;,&amp;quot;instanceArgs&amp;quot;:{&amp;quot;namespace&amp;quot;:&amp;quot;html&amp;quot;,&amp;quot;tag&amp;quot;:&amp;quot;div&amp;quot;},&amp;quot;children&amp;quot;:[]},{&amp;quot;props&amp;quot;:{&amp;quot;style&amp;quot;:{&amp;quot;height&amp;quot;:&amp;quot;0.0px&amp;quot;,&amp;quot;width&amp;quot;:&amp;quot;1.0em&amp;quot;}},&amp;quot;nodeType&amp;quot;:&amp;quot;DOM&amp;quot;,&amp;quot;type&amp;quot;:&amp;quot;node&amp;quot;,&amp;quot;instanceArgs&amp;quot;:{&amp;quot;namespace&amp;quot;:&amp;quot;html&amp;quot;,&amp;quot;tag&amp;quot;:&amp;quot;div&amp;quot;},&amp;quot;children&amp;quot;:[]},&amp;quot;label: 1&amp;quot;]}]})&lt;/unsafe-script&gt;&quot;},&quot;nodeType&quot;:&quot;DOM&quot;,&quot;type&quot;:&quot;node&quot;,&quot;instanceArgs&quot;:{&quot;namespace&quot;:&quot;html&quot;,&quot;tag&quot;:&quot;div&quot;},&quot;children&quot;:[]}]}]})</unsafe-script>\n",
       "</div>"
      ],
      "text/plain": [
       "Widget{:manipulate,Any}(OrderedDict{Symbol,Any}(:i=>Widget{:slider,Int64}(OrderedDict{Symbol,Any}(:changes=>Observable{Int64} with 1 listeners. Value:\n",
       "0,:value=>Observable{Int64} with 2 listeners. Value:\n",
       "30000), Observable{Int64} with 2 listeners. Value:\n",
       "30000, Scope(\"knockout-component-b0053a2f-bc3b-45b5-b337-359219f0ffa7\", Node{DOM}(DOM(:html, :div), Any[Node{DOM}(DOM(:html, :div), Any[Node{DOM}(DOM(:html, :label), Any[\"i\"], Dict{Symbol,Any}(:className=>\"interact \",:style=>Dict{Any,Any}(:padding=>\"5px 10px 0px 10px\")), 1)], Dict{Symbol,Any}(:attributes=>Dict(\"class\"=>\"interact-flex-row-left\")), 2), Node{DOM}(DOM(:html, :div), Any[Node{DOM}(DOM(:html, :input), Any[], Dict{Symbol,Any}(:max=>60000,:min=>1,:attributes=>Dict{Any,Any}(:type=>\"range\",Symbol(\"data-bind\")=>\"numericValue: value, valueUpdate: 'input', event: {change : function () {this.changes(this.changes()+1)}}\",\"orient\"=>\"horizontal\"),:step=>1,:className=>\"slider slider is-fullwidth\",:style=>Dict{Any,Any}()), 0)], Dict{Symbol,Any}(:attributes=>Dict(\"class\"=>\"interact-flex-row-center\")), 1), Node{DOM}(DOM(:html, :div), Any[Node{DOM}(DOM(:html, :p), Any[], Dict{Symbol,Any}(:attributes=>Dict(\"data-bind\"=>\"text: value\")), 0)], Dict{Symbol,Any}(:attributes=>Dict(\"class\"=>\"interact-flex-row-right\")), 1)], Dict{Symbol,Any}(:attributes=>Dict(\"class\"=>\"interact-flex-row\")), 7), Dict{String,Tuple{Observables.AbstractObservable,Union{Nothing, Bool}}}(\"changes\"=>(Observable{Int64} with 1 listeners. Value:\n",
       "0, nothing),\"value\"=>(Observable{Int64} with 2 listeners. Value:\n",
       "30000, nothing)), Set(String[]), nothing, Any[\"knockout\"=>\"/Users/alanedelman/.julia/packages/Knockout/JIqpG/src/../assets/knockout.js\", \"knockout_punches\"=>\"/Users/alanedelman/.julia/packages/Knockout/JIqpG/src/../assets/knockout_punches.js\", \"/Users/alanedelman/.julia/packages/InteractBase/PTCUD/src/../assets/all.js\", \"/Users/alanedelman/.julia/packages/InteractBase/PTCUD/src/../assets/style.css\", \"/Users/alanedelman/.julia/packages/InteractBulma/PH56C/src/../assets/main.css\"], Dict{Any,Any}(\"_promises\"=>Dict{Any,Any}(\"importsLoaded\"=>Any[JSString(\"function (ko, koPunches) {\\n    ko.punches.enableAll();\\n    ko.bindingHandlers.numericValue = {\\n        init : function(element, valueAccessor, allBindings, data, context) {\\n            var stringified = ko.observable(ko.unwrap(valueAccessor()));\\n            stringified.subscribe(function(value) {\\n                var val = parseFloat(value);\\n                if (!isNaN(val)) {\\n                    valueAccessor()(val);\\n                }\\n            })\\n            valueAccessor().subscribe(function(value) {\\n                var str = JSON.stringify(value);\\n                if ((str == \\\"0\\\") && ([\\\"-0\\\", \\\"-0.\\\"].indexOf(stringified()) >= 0))\\n                     return;\\n                 if ([\\\"null\\\", \\\"\\\"].indexOf(str) >= 0)\\n                     return;\\n                stringified(str);\\n            })\\n            ko.applyBindingsToNode(element, { value: stringified, valueUpdate: allBindings.get('valueUpdate')}, context);\\n        }\\n    };\\n    var json_data = JSON.parse(\\\"{\\\\\\\"changes\\\\\\\":0,\\\\\\\"value\\\\\\\":30000}\\\");\\n    var self = this;\\n    function AppViewModel() {\\n        for (var key in json_data) {\\n            var el = json_data[key];\\n            this[key] = Array.isArray(el) ? ko.observableArray(el) : ko.observable(el);\\n        }\\n        \\n        \\n        [this[\\\"changes\\\"].subscribe((function (val){!(this.valueFromJulia[\\\"changes\\\"]) ? (WebIO.setval({\\\"name\\\":\\\"changes\\\",\\\"scope\\\":\\\"knockout-component-b0053a2f-bc3b-45b5-b337-359219f0ffa7\\\",\\\"id\\\":\\\"ob_76\\\",\\\"type\\\":\\\"observable\\\"},val)) : undefined; return this.valueFromJulia[\\\"changes\\\"]=false}),self),this[\\\"value\\\"].subscribe((function (val){!(this.valueFromJulia[\\\"value\\\"]) ? (WebIO.setval({\\\"name\\\":\\\"value\\\",\\\"scope\\\":\\\"knockout-component-b0053a2f-bc3b-45b5-b337-359219f0ffa7\\\",\\\"id\\\":\\\"ob_75\\\",\\\"type\\\":\\\"observable\\\"},val)) : undefined; return this.valueFromJulia[\\\"value\\\"]=false}),self)]\\n        \\n    }\\n    self.model = new AppViewModel();\\n    self.valueFromJulia = {};\\n    for (var key in json_data) {\\n        self.valueFromJulia[key] = false;\\n    }\\n    ko.applyBindings(self.model, self.dom);\\n}\\n\")]),\"changes\"=>Any[JSString(\"(function (val){return (val!=this.model[\\\"changes\\\"]()) ? (this.valueFromJulia[\\\"changes\\\"]=true, this.model[\\\"changes\\\"](val)) : undefined})\")],\"value\"=>Any[JSString(\"(function (val){return (val!=this.model[\\\"value\\\"]()) ? (this.valueFromJulia[\\\"value\\\"]=true, this.model[\\\"value\\\"](val)) : undefined})\")]), ConnectionPool(Channel{Any}(sz_max:9223372036854775807,sz_curr:3), Set(AbstractConnection[]), Channel{AbstractConnection}(sz_max:32,sz_curr:0))), ##52#53{#dom#15{##dom#13#14{Dict{Any,Any},DOM}},typeof(scope)}(#dom#15{##dom#13#14{Dict{Any,Any},DOM}}(##dom#13#14{Dict{Any,Any},DOM}(Dict{Any,Any}(:className=>\"field\"), DOM(:html, :div))), scope))), Observable{Any} with 0 listeners. Value:\n",
       "Node{WebIO.DOM}(WebIO.DOM(:html, :div), Any[Node{DOM}(DOM(:html, :div), Any[], Dict{Symbol,Any}(:setInnerHtml=>\"<img src='data:image/png;base64,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'></img>\"), 0), Node{DOM}(DOM(:html, :div), Any[], Dict{Symbol,Any}(:style=>Dict{String,Measures.Length{U,Float64} where U}(\"height\"=>0.0px,\"width\"=>1.0em)), 0), \"label: 1\"], Dict{Symbol,Any}(:style=>Dict(\"display\"=>\"flex\",\"flex-direction\"=>\"row\")), 3), nothing, getfield(InteractBase, Symbol(\"##142#143\"))())"
      ]
     },
     "execution_count": 109,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "@manipulate for i in eachindex(images)\n",
    "    hbox(images[i], hskip(1em), string(\"label: \", labels[i]))\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## An image is just data"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 110,
   "metadata": {},
   "outputs": [],
   "source": [
    "a = vec(images[2][10, :]);"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 111,
   "metadata": {},
   "outputs": [],
   "source": [
    "b = round.(Int, 256*images[2][10, i].val for i in 1:28);"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 112,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/svg+xml": [
       "<?xml version=\"1.0\" encoding=\"UTF-8\"?>\n",
       "<!DOCTYPE svg PUBLIC \"-//W3C//DTD SVG 1.1//EN\"\n",
       " \"http://www.w3.org/Graphics/SVG/1.1/DTD/svg11.dtd\">\n",
       "<svg xmlns=\"http://www.w3.org/2000/svg\" version=\"1.1\"\n",
       "     width=\"25mm\" height=\"25mm\" viewBox=\"0 0 1 1\">\n",
       "     <rect width=\"1\" height=\"1\"\n",
       "           fill=\"#000000\" stroke=\"none\"/>\n",
       "</svg>\n"
      ],
      "text/plain": [
       "Gray{N0f8}(0.0)"
      ]
     },
     "execution_count": 112,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a[1]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 113,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0"
      ]
     },
     "execution_count": 113,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "b[1]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 114,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "\u001b[32m\u001b[1m Resolving\u001b[22m\u001b[39m package versions...\n",
      "\u001b[32m\u001b[1m  Updating\u001b[22m\u001b[39m `~/.julia/environments/v1.0/Project.toml`\n",
      "\u001b[90m [no changes]\u001b[39m\n",
      "\u001b[32m\u001b[1m  Updating\u001b[22m\u001b[39m `~/.julia/environments/v1.0/Manifest.toml`\n",
      "\u001b[90m [no changes]\u001b[39m\n"
     ]
    }
   ],
   "source": [
    "using Pkg\n",
    "Pkg.add(\"Interact\")\n",
    "using Interact"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 115,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<div class='tex2jax_ignore interactbulma'>\n",
       "<div class='display:none'></div><unsafe-script style='display:none'>\n",
       "WebIO.mount(this.previousSibling,{&quot;props&quot;:{},&quot;nodeType&quot;:&quot;DOM&quot;,&quot;type&quot;:&quot;node&quot;,&quot;instanceArgs&quot;:{&quot;namespace&quot;:&quot;html&quot;,&quot;tag&quot;:&quot;div&quot;},&quot;children&quot;:[{&quot;props&quot;:{&quot;className&quot;:&quot;field&quot;},&quot;nodeType&quot;:&quot;DOM&quot;,&quot;type&quot;:&quot;node&quot;,&quot;instanceArgs&quot;:{&quot;namespace&quot;:&quot;html&quot;,&quot;tag&quot;:&quot;div&quot;},&quot;children&quot;:[{&quot;props&quot;:{},&quot;nodeType&quot;:&quot;Scope&quot;,&quot;type&quot;:&quot;node&quot;,&quot;instanceArgs&quot;:{&quot;imports&quot;:{&quot;data&quot;:[{&quot;name&quot;:&quot;knockout&quot;,&quot;type&quot;:&quot;js&quot;,&quot;url&quot;:&quot;/assetserver/06c67466ebc6c52906ffd4101b7c74a47f9567bc-knockout.js&quot;},{&quot;name&quot;:&quot;knockout_punches&quot;,&quot;type&quot;:&quot;js&quot;,&quot;url&quot;:&quot;/assetserver/46b0ee2b2af2730ca29ddb46f091ac108fb8f609-knockout_punches.js&quot;},{&quot;name&quot;:null,&quot;type&quot;:&quot;js&quot;,&quot;url&quot;:&quot;/assetserver/e923a2cf7cc8bb183f572d980ae606c5f08fa0dc-all.js&quot;},{&quot;name&quot;:null,&quot;type&quot;:&quot;css&quot;,&quot;url&quot;:&quot;/assetserver/b973c333a7d1e5a6ca74f1442a2f26b5e4d0580e-style.css&quot;},{&quot;name&quot;:null,&quot;type&quot;:&quot;css&quot;,&quot;url&quot;:&quot;/assetserver/885f9c98e35e2ecc817c654a867b4570a606ec22-main.css&quot;}],&quot;type&quot;:&quot;async_block&quot;},&quot;id&quot;:&quot;knockout-component-c2e666e1-b9bd-4c53-a776-7256a6d7979d&quot;,&quot;handlers&quot;:{&quot;_promises&quot;:{&quot;importsLoaded&quot;:[function (ko, koPunches) {\n",
       "    ko.punches.enableAll();\n",
       "    ko.bindingHandlers.numericValue = {\n",
       "        init : function(element, valueAccessor, allBindings, data, context) {\n",
       "            var stringified = ko.observable(ko.unwrap(valueAccessor()));\n",
       "            stringified.subscribe(function(value) {\n",
       "                var val = parseFloat(value);\n",
       "                if (!isNaN(val)) {\n",
       "                    valueAccessor()(val);\n",
       "                }\n",
       "            })\n",
       "            valueAccessor().subscribe(function(value) {\n",
       "                var str = JSON.stringify(value);\n",
       "                if ((str == &quot;0&quot;) &amp;&amp; ([&quot;-0&quot;, &quot;-0.&quot;].indexOf(stringified()) &gt;= 0))\n",
       "                     return;\n",
       "                 if ([&quot;null&quot;, &quot;&quot;].indexOf(str) &gt;= 0)\n",
       "                     return;\n",
       "                stringified(str);\n",
       "            })\n",
       "            ko.applyBindingsToNode(element, { value: stringified, valueUpdate: allBindings.get(&#39;valueUpdate&#39;)}, context);\n",
       "        }\n",
       "    };\n",
       "    var json_data = JSON.parse(&quot;{\\&quot;changes\\&quot;:0,\\&quot;value\\&quot;:14}&quot;);\n",
       "    var self = this;\n",
       "    function AppViewModel() {\n",
       "        for (var key in json_data) {\n",
       "            var el = json_data[key];\n",
       "            this[key] = Array.isArray(el) ? ko.observableArray(el) : ko.observable(el);\n",
       "        }\n",
       "        \n",
       "        \n",
       "        [this[&quot;changes&quot;].subscribe((function (val){!(this.valueFromJulia[&quot;changes&quot;]) ? (WebIO.setval({&quot;name&quot;:&quot;changes&quot;,&quot;scope&quot;:&quot;knockout-component-c2e666e1-b9bd-4c53-a776-7256a6d7979d&quot;,&quot;id&quot;:&quot;ob_81&quot;,&quot;type&quot;:&quot;observable&quot;},val)) : undefined; return this.valueFromJulia[&quot;changes&quot;]=false}),self),this[&quot;value&quot;].subscribe((function (val){!(this.valueFromJulia[&quot;value&quot;]) ? (WebIO.setval({&quot;name&quot;:&quot;value&quot;,&quot;scope&quot;:&quot;knockout-component-c2e666e1-b9bd-4c53-a776-7256a6d7979d&quot;,&quot;id&quot;:&quot;ob_80&quot;,&quot;type&quot;:&quot;observable&quot;},val)) : undefined; return this.valueFromJulia[&quot;value&quot;]=false}),self)]\n",
       "        \n",
       "    }\n",
       "    self.model = new AppViewModel();\n",
       "    self.valueFromJulia = {};\n",
       "    for (var key in json_data) {\n",
       "        self.valueFromJulia[key] = false;\n",
       "    }\n",
       "    ko.applyBindings(self.model, self.dom);\n",
       "}\n",
       "]},&quot;changes&quot;:[(function (val){return (val!=this.model[&quot;changes&quot;]()) ? (this.valueFromJulia[&quot;changes&quot;]=true, this.model[&quot;changes&quot;](val)) : undefined})],&quot;value&quot;:[(function (val){return (val!=this.model[&quot;value&quot;]()) ? (this.valueFromJulia[&quot;value&quot;]=true, this.model[&quot;value&quot;](val)) : undefined})]},&quot;systemjs_options&quot;:null,&quot;observables&quot;:{&quot;changes&quot;:{&quot;sync&quot;:false,&quot;id&quot;:&quot;ob_81&quot;,&quot;value&quot;:0},&quot;value&quot;:{&quot;sync&quot;:true,&quot;id&quot;:&quot;ob_80&quot;,&quot;value&quot;:14}}},&quot;children&quot;:[{&quot;props&quot;:{&quot;attributes&quot;:{&quot;class&quot;:&quot;interact-flex-row&quot;}},&quot;nodeType&quot;:&quot;DOM&quot;,&quot;type&quot;:&quot;node&quot;,&quot;instanceArgs&quot;:{&quot;namespace&quot;:&quot;html&quot;,&quot;tag&quot;:&quot;div&quot;},&quot;children&quot;:[{&quot;props&quot;:{&quot;attributes&quot;:{&quot;class&quot;:&quot;interact-flex-row-left&quot;}},&quot;nodeType&quot;:&quot;DOM&quot;,&quot;type&quot;:&quot;node&quot;,&quot;instanceArgs&quot;:{&quot;namespace&quot;:&quot;html&quot;,&quot;tag&quot;:&quot;div&quot;},&quot;children&quot;:[{&quot;props&quot;:{&quot;className&quot;:&quot;interact &quot;,&quot;style&quot;:{&quot;padding&quot;:&quot;5px 10px 0px 10px&quot;}},&quot;nodeType&quot;:&quot;DOM&quot;,&quot;type&quot;:&quot;node&quot;,&quot;instanceArgs&quot;:{&quot;namespace&quot;:&quot;html&quot;,&quot;tag&quot;:&quot;label&quot;},&quot;children&quot;:[&quot;j&quot;]}]},{&quot;props&quot;:{&quot;attributes&quot;:{&quot;class&quot;:&quot;interact-flex-row-center&quot;}},&quot;nodeType&quot;:&quot;DOM&quot;,&quot;type&quot;:&quot;node&quot;,&quot;instanceArgs&quot;:{&quot;namespace&quot;:&quot;html&quot;,&quot;tag&quot;:&quot;div&quot;},&quot;children&quot;:[{&quot;props&quot;:{&quot;max&quot;:28,&quot;min&quot;:1,&quot;attributes&quot;:{&quot;type&quot;:&quot;range&quot;,&quot;data-bind&quot;:&quot;numericValue: value, valueUpdate: &#39;input&#39;, event: {change : function () {this.changes(this.changes()+1)}}&quot;,&quot;orient&quot;:&quot;horizontal&quot;},&quot;step&quot;:1,&quot;className&quot;:&quot;slider slider is-fullwidth&quot;,&quot;style&quot;:{}},&quot;nodeType&quot;:&quot;DOM&quot;,&quot;type&quot;:&quot;node&quot;,&quot;instanceArgs&quot;:{&quot;namespace&quot;:&quot;html&quot;,&quot;tag&quot;:&quot;input&quot;},&quot;children&quot;:[]}]},{&quot;props&quot;:{&quot;attributes&quot;:{&quot;class&quot;:&quot;interact-flex-row-right&quot;}},&quot;nodeType&quot;:&quot;DOM&quot;,&quot;type&quot;:&quot;node&quot;,&quot;instanceArgs&quot;:{&quot;namespace&quot;:&quot;html&quot;,&quot;tag&quot;:&quot;div&quot;},&quot;children&quot;:[{&quot;props&quot;:{&quot;attributes&quot;:{&quot;data-bind&quot;:&quot;text: value&quot;}},&quot;nodeType&quot;:&quot;DOM&quot;,&quot;type&quot;:&quot;node&quot;,&quot;instanceArgs&quot;:{&quot;namespace&quot;:&quot;html&quot;,&quot;tag&quot;:&quot;p&quot;},&quot;children&quot;:[]}]}]}]}]},{&quot;props&quot;:{},&quot;nodeType&quot;:&quot;Scope&quot;,&quot;type&quot;:&quot;node&quot;,&quot;instanceArgs&quot;:{&quot;imports&quot;:{&quot;data&quot;:[],&quot;type&quot;:&quot;async_block&quot;},&quot;id&quot;:&quot;scope-e37e71de-6e6d-4767-8be0-c733506dbdaa&quot;,&quot;handlers&quot;:{&quot;obs-output&quot;:[function (updated_htmlstr) {\n",
       "    var el = this.dom.querySelector(&quot;#out&quot;);\n",
       "    WebIO.propUtils.setInnerHtml(el, updated_htmlstr);\n",
       "}]},&quot;systemjs_options&quot;:null,&quot;observables&quot;:{&quot;obs-output&quot;:{&quot;sync&quot;:false,&quot;id&quot;:&quot;ob_84&quot;,&quot;value&quot;:&quot;&lt;div class=&#39;display:none&#39;&gt;&lt;/div&gt;&lt;unsafe-script style=&#39;display:none&#39;&gt;\\nWebIO.mount(this.previousSibling,{&amp;quot;props&amp;quot;:{&amp;quot;attributes&amp;quot;:{&amp;quot;class&amp;quot;:&amp;quot;interact-flex-row&amp;quot;}},&amp;quot;nodeType&amp;quot;:&amp;quot;DOM&amp;quot;,&amp;quot;type&amp;quot;:&amp;quot;node&amp;quot;,&amp;quot;instanceArgs&amp;quot;:{&amp;quot;namespace&amp;quot;:&amp;quot;html&amp;quot;,&amp;quot;tag&amp;quot;:&amp;quot;div&amp;quot;},&amp;quot;children&amp;quot;:[{&amp;quot;props&amp;quot;:{&amp;quot;style&amp;quot;:{&amp;quot;display&amp;quot;:&amp;quot;flex&amp;quot;,&amp;quot;flex-direction&amp;quot;:&amp;quot;row&amp;quot;}},&amp;quot;nodeType&amp;quot;:&amp;quot;DOM&amp;quot;,&amp;quot;type&amp;quot;:&amp;quot;node&amp;quot;,&amp;quot;instanceArgs&amp;quot;:{&amp;quot;namespace&amp;quot;:&amp;quot;html&amp;quot;,&amp;quot;tag&amp;quot;:&amp;quot;div&amp;quot;},&amp;quot;children&amp;quot;:[{&amp;quot;props&amp;quot;:{&amp;quot;style&amp;quot;:{&amp;quot;display&amp;quot;:&amp;quot;flex&amp;quot;,&amp;quot;flex-direction&amp;quot;:&amp;quot;column&amp;quot;}},&amp;quot;nodeType&amp;quot;:&amp;quot;DOM&amp;quot;,&amp;quot;type&amp;quot;:&amp;quot;node&amp;quot;,&amp;quot;instanceArgs&amp;quot;:{&amp;quot;namespace&amp;quot;:&amp;quot;html&amp;quot;,&amp;quot;tag&amp;quot;:&amp;quot;div&amp;quot;},&amp;quot;children&amp;quot;:[{&amp;quot;props&amp;quot;:{&amp;quot;setInnerHtml&amp;quot;:&amp;quot;&amp;lt;?xml version=\\\\&amp;quot;1.0\\\\&amp;quot; encoding=\\\\&amp;quot;UTF-8\\\\&amp;quot;?&amp;gt;\\\\n&amp;lt;!DOCTYPE svg PUBLIC \\\\&amp;quot;-//W3C//DTD SVG 1.1//EN\\\\&amp;quot;\\\\n \\\\&amp;quot;http://www.w3.org/Graphics/SVG/1.1/DTD/svg11.dtd\\\\&amp;quot;&amp;gt;\\\\n&amp;lt;svg xmlns=\\\\&amp;quot;http://www.w3.org/2000/svg\\\\&amp;quot; version=\\\\&amp;quot;1.1\\\\&amp;quot;\\\\n     width=\\\\&amp;quot;25mm\\\\&amp;quot; height=\\\\&amp;quot;25mm\\\\&amp;quot; viewBox=\\\\&amp;quot;0 0 1 1\\\\&amp;quot;&amp;gt;\\\\n     &amp;lt;rect width=\\\\&amp;quot;1\\\\&amp;quot; height=\\\\&amp;quot;1\\\\&amp;quot;\\\\n           fill=\\\\&amp;quot;#000000\\\\&amp;quot; stroke=\\\\&amp;quot;none\\\\&amp;quot;/&amp;gt;\\\\n&amp;lt;/svg&amp;gt;\\\\n&amp;quot;},&amp;quot;nodeType&amp;quot;:&amp;quot;DOM&amp;quot;,&amp;quot;type&amp;quot;:&amp;quot;node&amp;quot;,&amp;quot;instanceArgs&amp;quot;:{&amp;quot;namespace&amp;quot;:&amp;quot;html&amp;quot;,&amp;quot;tag&amp;quot;:&amp;quot;div&amp;quot;},&amp;quot;children&amp;quot;:[]},&amp;quot; 0 &amp;quot;]},{&amp;quot;props&amp;quot;:{&amp;quot;style&amp;quot;:{&amp;quot;display&amp;quot;:&amp;quot;flex&amp;quot;,&amp;quot;flex-direction&amp;quot;:&amp;quot;column&amp;quot;}},&amp;quot;nodeType&amp;quot;:&amp;quot;DOM&amp;quot;,&amp;quot;type&amp;quot;:&amp;quot;node&amp;quot;,&amp;quot;instanceArgs&amp;quot;:{&amp;quot;namespace&amp;quot;:&amp;quot;html&amp;quot;,&amp;quot;tag&amp;quot;:&amp;quot;div&amp;quot;},&amp;quot;children&amp;quot;:[{&amp;quot;props&amp;quot;:{&amp;quot;setInnerHtml&amp;quot;:&amp;quot;&amp;lt;?xml version=\\\\&amp;quot;1.0\\\\&amp;quot; encoding=\\\\&amp;quot;UTF-8\\\\&amp;quot;?&amp;gt;\\\\n&amp;lt;!DOCTYPE svg PUBLIC \\\\&amp;quot;-//W3C//DTD SVG 1.1//EN\\\\&amp;quot;\\\\n \\\\&amp;quot;http://www.w3.org/Graphics/SVG/1.1/DTD/svg11.dtd\\\\&amp;quot;&amp;gt;\\\\n&amp;lt;svg xmlns=\\\\&amp;quot;http://www.w3.org/2000/svg\\\\&amp;quot; version=\\\\&amp;quot;1.1\\\\&amp;quot;\\\\n     width=\\\\&amp;quot;25mm\\\\&amp;quot; height=\\\\&amp;quot;25mm\\\\&amp;quot; viewBox=\\\\&amp;quot;0 0 1 1\\\\&amp;quot;&amp;gt;\\\\n     &amp;lt;rect width=\\\\&amp;quot;1\\\\&amp;quot; height=\\\\&amp;quot;1\\\\&amp;quot;\\\\n           fill=\\\\&amp;quot;#000000\\\\&amp;quot; stroke=\\\\&amp;quot;none\\\\&amp;quot;/&amp;gt;\\\\n&amp;lt;/svg&amp;gt;\\\\n&amp;quot;},&amp;quot;nodeType&amp;quot;:&amp;quot;DOM&amp;quot;,&amp;quot;type&amp;quot;:&amp;quot;node&amp;quot;,&amp;quot;instanceArgs&amp;quot;:{&amp;quot;namespace&amp;quot;:&amp;quot;html&amp;quot;,&amp;quot;tag&amp;quot;:&amp;quot;div&amp;quot;},&amp;quot;children&amp;quot;:[]},&amp;quot; 0 &amp;quot;]},{&amp;quot;props&amp;quot;:{&amp;quot;style&amp;quot;:{&amp;quot;display&amp;quot;:&amp;quot;flex&amp;quot;,&amp;quot;flex-direction&amp;quot;:&amp;quot;column&amp;quot;}},&amp;quot;nodeType&amp;quot;:&amp;quot;DOM&amp;quot;,&amp;quot;type&amp;quot;:&amp;quot;node&amp;quot;,&amp;quot;instanceArgs&amp;quot;:{&amp;quot;namespace&amp;quot;:&amp;quot;html&amp;quot;,&amp;quot;tag&amp;quot;:&amp;quot;div&amp;quot;},&amp;quot;children&amp;quot;:[{&amp;quot;props&amp;quot;:{&amp;quot;setInnerHtml&amp;quot;:&amp;quot;&amp;lt;?xml version=\\\\&amp;quot;1.0\\\\&amp;quot; encoding=\\\\&amp;quot;UTF-8\\\\&amp;quot;?&amp;gt;\\\\n&amp;lt;!DOCTYPE svg PUBLIC \\\\&amp;quot;-//W3C//DTD SVG 1.1//EN\\\\&amp;quot;\\\\n \\\\&amp;quot;http://www.w3.org/Graphics/SVG/1.1/DTD/svg11.dtd\\\\&amp;quot;&amp;gt;\\\\n&amp;lt;svg xmlns=\\\\&amp;quot;http://www.w3.org/2000/svg\\\\&amp;quot; version=\\\\&amp;quot;1.1\\\\&amp;quot;\\\\n     width=\\\\&amp;quot;25mm\\\\&amp;quot; height=\\\\&amp;quot;25mm\\\\&amp;quot; viewBox=\\\\&amp;quot;0 0 1 1\\\\&amp;quot;&amp;gt;\\\\n     &amp;lt;rect width=\\\\&amp;quot;1\\\\&amp;quot; height=\\\\&amp;quot;1\\\\&amp;quot;\\\\n           fill=\\\\&amp;quot;#000000\\\\&amp;quot; stroke=\\\\&amp;quot;none\\\\&amp;quot;/&amp;gt;\\\\n&amp;lt;/svg&amp;gt;\\\\n&amp;quot;},&amp;quot;nodeType&amp;quot;:&amp;quot;DOM&amp;quot;,&amp;quot;type&amp;quot;:&amp;quot;node&amp;quot;,&amp;quot;instanceArgs&amp;quot;:{&amp;quot;namespace&amp;quot;:&amp;quot;html&amp;quot;,&amp;quot;tag&amp;quot;:&amp;quot;div&amp;quot;},&amp;quot;children&amp;quot;:[]},&amp;quot; 0 &amp;quot;]},{&amp;quot;props&amp;quot;:{&amp;quot;style&amp;quot;:{&amp;quot;display&amp;quot;:&amp;quot;flex&amp;quot;,&amp;quot;flex-direction&amp;quot;:&amp;quot;column&amp;quot;}},&amp;quot;nodeType&amp;quot;:&amp;quot;DOM&amp;quot;,&amp;quot;type&amp;quot;:&amp;quot;node&amp;quot;,&amp;quot;instanceArgs&amp;quot;:{&amp;quot;namespace&amp;quot;:&amp;quot;html&amp;quot;,&amp;quot;tag&amp;quot;:&amp;quot;div&amp;quot;},&amp;quot;children&amp;quot;:[{&amp;quot;props&amp;quot;:{&amp;quot;setInnerHtml&amp;quot;:&amp;quot;&amp;lt;?xml version=\\\\&amp;quot;1.0\\\\&amp;quot; encoding=\\\\&amp;quot;UTF-8\\\\&amp;quot;?&amp;gt;\\\\n&amp;lt;!DOCTYPE svg PUBLIC \\\\&amp;quot;-//W3C//DTD SVG 1.1//EN\\\\&amp;quot;\\\\n \\\\&amp;quot;http://www.w3.org/Graphics/SVG/1.1/DTD/svg11.dtd\\\\&amp;quot;&amp;gt;\\\\n&amp;lt;svg xmlns=\\\\&amp;quot;http://www.w3.org/2000/svg\\\\&amp;quot; version=\\\\&amp;quot;1.1\\\\&amp;quot;\\\\n     width=\\\\&amp;quot;25mm\\\\&amp;quot; height=\\\\&amp;quot;25mm\\\\&amp;quot; viewBox=\\\\&amp;quot;0 0 1 1\\\\&amp;quot;&amp;gt;\\\\n     &amp;lt;rect width=\\\\&amp;quot;1\\\\&amp;quot; height=\\\\&amp;quot;1\\\\&amp;quot;\\\\n           fill=\\\\&amp;quot;#000000\\\\&amp;quot; stroke=\\\\&amp;quot;none\\\\&amp;quot;/&amp;gt;\\\\n&amp;lt;/svg&amp;gt;\\\\n&amp;quot;},&amp;quot;nodeType&amp;quot;:&amp;quot;DOM&amp;quot;,&amp;quot;type&amp;quot;:&amp;quot;node&amp;quot;,&amp;quot;instanceArgs&amp;quot;:{&amp;quot;namespace&amp;quot;:&amp;quot;html&amp;quot;,&amp;quot;tag&amp;quot;:&amp;quot;div&amp;quot;},&amp;quot;children&amp;quot;:[]},&amp;quot; 0 &amp;quot;]},{&amp;quot;props&amp;quot;:{&amp;quot;style&amp;quot;:{&amp;quot;display&amp;quot;:&amp;quot;flex&amp;quot;,&amp;quot;flex-direction&amp;quot;:&amp;quot;column&amp;quot;}},&amp;quot;nodeType&amp;quot;:&amp;quot;DOM&amp;quot;,&amp;quot;type&amp;quot;:&amp;quot;node&amp;quot;,&amp;quot;instanceArgs&amp;quot;:{&amp;quot;namespace&amp;quot;:&amp;quot;html&amp;quot;,&amp;quot;tag&amp;quot;:&amp;quot;div&amp;quot;},&amp;quot;children&amp;quot;:[{&amp;quot;props&amp;quot;:{&amp;quot;setInnerHtml&amp;quot;:&amp;quot;&amp;lt;?xml version=\\\\&amp;quot;1.0\\\\&amp;quot; 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Node{DOM}(DOM(:html, :div), Any[Node{DOM}(DOM(:html, :div), Any[], Dict{Symbol,Any}(:setInnerHtml=>\"<?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?>\\n<!DOCTYPE svg PUBLIC \\\"-//W3C//DTD SVG 1.1//EN\\\"\\n \\\"http://www.w3.org/Graphics/SVG/1.1/DTD/svg11.dtd\\\">\\n<svg xmlns=\\\"http://www.w3.org/2000/svg\\\" version=\\\"1.1\\\"\\n     width=\\\"25mm\\\" height=\\\"25mm\\\" viewBox=\\\"0 0 1 1\\\">\\n     <rect width=\\\"1\\\" height=\\\"1\\\"\\n           fill=\\\"#C3C3C3\\\" stroke=\\\"none\\\"/>\\n</svg>\\n\"), 0), \" 196 \"], Dict{Symbol,Any}(:style=>Dict(\"display\"=>\"flex\",\"flex-direction\"=>\"column\")), 2), Node{DOM}(DOM(:html, :div), Any[Node{DOM}(DOM(:html, :div), Any[], Dict{Symbol,Any}(:setInnerHtml=>\"<?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?>\\n<!DOCTYPE svg PUBLIC \\\"-//W3C//DTD SVG 1.1//EN\\\"\\n \\\"http://www.w3.org/Graphics/SVG/1.1/DTD/svg11.dtd\\\">\\n<svg xmlns=\\\"http://www.w3.org/2000/svg\\\" version=\\\"1.1\\\"\\n     width=\\\"25mm\\\" height=\\\"25mm\\\" viewBox=\\\"0 0 1 1\\\">\\n     <rect width=\\\"1\\\" height=\\\"1\\\"\\n           fill=\\\"#000000\\\" stroke=\\\"none\\\"/>\\n</svg>\\n\"), 0), \" 0 \"], Dict{Symbol,Any}(:style=>Dict(\"display\"=>\"flex\",\"flex-direction\"=>\"column\")), 2), Node{DOM}(DOM(:html, :div), Any[Node{DOM}(DOM(:html, :div), Any[], Dict{Symbol,Any}(:setInnerHtml=>\"<?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?>\\n<!DOCTYPE svg PUBLIC \\\"-//W3C//DTD SVG 1.1//EN\\\"\\n \\\"http://www.w3.org/Graphics/SVG/1.1/DTD/svg11.dtd\\\">\\n<svg xmlns=\\\"http://www.w3.org/2000/svg\\\" version=\\\"1.1\\\"\\n     width=\\\"25mm\\\" height=\\\"25mm\\\" viewBox=\\\"0 0 1 1\\\">\\n     <rect width=\\\"1\\\" height=\\\"1\\\"\\n           fill=\\\"#000000\\\" stroke=\\\"none\\\"/>\\n</svg>\\n\"), 0), \" 0 \"], Dict{Symbol,Any}(:style=>Dict(\"display\"=>\"flex\",\"flex-direction\"=>\"column\")), 2), Node{DOM}(DOM(:html, :div), Any[Node{DOM}(DOM(:html, :div), Any[], Dict{Symbol,Any}(:setInnerHtml=>\"<?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?>\\n<!DOCTYPE svg PUBLIC \\\"-//W3C//DTD SVG 1.1//EN\\\"\\n \\\"http://www.w3.org/Graphics/SVG/1.1/DTD/svg11.dtd\\\">\\n<svg xmlns=\\\"http://www.w3.org/2000/svg\\\" version=\\\"1.1\\\"\\n     width=\\\"25mm\\\" height=\\\"25mm\\\" viewBox=\\\"0 0 1 1\\\">\\n     <rect width=\\\"1\\\" height=\\\"1\\\"\\n           fill=\\\"#000000\\\" stroke=\\\"none\\\"/>\\n</svg>\\n\"), 0), \" 0 \"], Dict{Symbol,Any}(:style=>Dict(\"display\"=>\"flex\",\"flex-direction\"=>\"column\")), 2), Node{DOM}(DOM(:html, :div), Any[Node{DOM}(DOM(:html, :div), Any[], Dict{Symbol,Any}(:setInnerHtml=>\"<?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?>\\n<!DOCTYPE svg PUBLIC \\\"-//W3C//DTD SVG 1.1//EN\\\"\\n \\\"http://www.w3.org/Graphics/SVG/1.1/DTD/svg11.dtd\\\">\\n<svg xmlns=\\\"http://www.w3.org/2000/svg\\\" version=\\\"1.1\\\"\\n     width=\\\"25mm\\\" height=\\\"25mm\\\" viewBox=\\\"0 0 1 1\\\">\\n     <rect width=\\\"1\\\" height=\\\"1\\\"\\n           fill=\\\"#000000\\\" stroke=\\\"none\\\"/>\\n</svg>\\n\"), 0), \" 0 \"], Dict{Symbol,Any}(:style=>Dict(\"display\"=>\"flex\",\"flex-direction\"=>\"column\")), 2), Node{DOM}(DOM(:html, :div), Any[Node{DOM}(DOM(:html, :div), Any[], Dict{Symbol,Any}(:setInnerHtml=>\"<?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?>\\n<!DOCTYPE svg PUBLIC \\\"-//W3C//DTD SVG 1.1//EN\\\"\\n \\\"http://www.w3.org/Graphics/SVG/1.1/DTD/svg11.dtd\\\">\\n<svg xmlns=\\\"http://www.w3.org/2000/svg\\\" version=\\\"1.1\\\"\\n     width=\\\"25mm\\\" height=\\\"25mm\\\" viewBox=\\\"0 0 1 1\\\">\\n     <rect width=\\\"1\\\" height=\\\"1\\\"\\n           fill=\\\"#000000\\\" stroke=\\\"none\\\"/>\\n</svg>\\n\"), 0), \" 0 \"], Dict{Symbol,Any}(:style=>Dict(\"display\"=>\"flex\",\"flex-direction\"=>\"column\")), 2)], Dict{Symbol,Any}(:style=>Dict(\"display\"=>\"flex\",\"flex-direction\"=>\"row\")), 84), nothing, getfield(InteractBase, Symbol(\"##142#143\"))())"
      ]
     },
     "execution_count": 115,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "@manipulate for j in 1:28\n",
    "    a = vec(images[2][j, :])\n",
    "    b = round.(Int, 256*images[2][j, i].val for i in 1:28);\n",
    "    hbox([vbox(a[i], string(\" \", b[i], \" \" )) for i in 1:28])\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 48,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<div class='tex2jax_ignore interactbulma'>\n",
       "<div class='display:none'></div><unsafe-script style='display:none'>\n",
       "WebIO.mount(this.previousSibling,{&quot;props&quot;:{},&quot;nodeType&quot;:&quot;DOM&quot;,&quot;type&quot;:&quot;node&quot;,&quot;instanceArgs&quot;:{&quot;namespace&quot;:&quot;html&quot;,&quot;tag&quot;:&quot;div&quot;},&quot;children&quot;:[{&quot;props&quot;:{&quot;className&quot;:&quot;field&quot;},&quot;nodeType&quot;:&quot;DOM&quot;,&quot;type&quot;:&quot;node&quot;,&quot;instanceArgs&quot;:{&quot;namespace&quot;:&quot;html&quot;,&quot;tag&quot;:&quot;div&quot;},&quot;children&quot;:[{&quot;props&quot;:{},&quot;nodeType&quot;:&quot;Scope&quot;,&quot;type&quot;:&quot;node&quot;,&quot;instanceArgs&quot;:{&quot;imports&quot;:{&quot;data&quot;:[{&quot;name&quot;:&quot;knockout&quot;,&quot;type&quot;:&quot;js&quot;,&quot;url&quot;:&quot;/assetserver/06c67466ebc6c52906ffd4101b7c74a47f9567bc-knockout.js&quot;},{&quot;name&quot;:&quot;knockout_punches&quot;,&quot;type&quot;:&quot;js&quot;,&quot;url&quot;:&quot;/assetserver/46b0ee2b2af2730ca29ddb46f091ac108fb8f609-knockout_punches.js&quot;},{&quot;name&quot;:null,&quot;type&quot;:&quot;js&quot;,&quot;url&quot;:&quot;/assetserver/e923a2cf7cc8bb183f572d980ae606c5f08fa0dc-all.js&quot;},{&quot;name&quot;:null,&quot;type&quot;:&quot;css&quot;,&quot;url&quot;:&quot;/assetserver/b973c333a7d1e5a6ca74f1442a2f26b5e4d0580e-style.css&quot;},{&quot;name&quot;:null,&quot;type&quot;:&quot;css&quot;,&quot;url&quot;:&quot;/assetserver/885f9c98e35e2ecc817c654a867b4570a606ec22-main.css&quot;}],&quot;type&quot;:&quot;async_block&quot;},&quot;id&quot;:&quot;knockout-component-fbef08bd-21d6-4ec3-b3b9-28989f3a311d&quot;,&quot;handlers&quot;:{&quot;_promises&quot;:{&quot;importsLoaded&quot;:[function (ko, koPunches) {\n",
       "    ko.punches.enableAll();\n",
       "    ko.bindingHandlers.numericValue = {\n",
       "        init : function(element, valueAccessor, allBindings, data, context) {\n",
       "            var stringified = ko.observable(ko.unwrap(valueAccessor()));\n",
       "            stringified.subscribe(function(value) {\n",
       "                var val = parseFloat(value);\n",
       "                if (!isNaN(val)) {\n",
       "                    valueAccessor()(val);\n",
       "                }\n",
       "            })\n",
       "            valueAccessor().subscribe(function(value) {\n",
       "                var str = JSON.stringify(value);\n",
       "                if ((str == &quot;0&quot;) &amp;&amp; ([&quot;-0&quot;, &quot;-0.&quot;].indexOf(stringified()) &gt;= 0))\n",
       "                     return;\n",
       "                 if ([&quot;null&quot;, &quot;&quot;].indexOf(str) &gt;= 0)\n",
       "                     return;\n",
       "                stringified(str);\n",
       "            })\n",
       "            ko.applyBindingsToNode(element, { value: stringified, valueUpdate: allBindings.get(&#39;valueUpdate&#39;)}, context);\n",
       "        }\n",
       "    };\n",
       "    var json_data = JSON.parse(&quot;{\\&quot;changes\\&quot;:0,\\&quot;value\\&quot;:14}&quot;);\n",
       "    var self = this;\n",
       "    function AppViewModel() {\n",
       "        for (var key in json_data) {\n",
       "            var el = json_data[key];\n",
       "            this[key] = Array.isArray(el) ? ko.observableArray(el) : ko.observable(el);\n",
       "        }\n",
       "        \n",
       "        \n",
       "        [this[&quot;changes&quot;].subscribe((function (val){!(this.valueFromJulia[&quot;changes&quot;]) ? (WebIO.setval({&quot;name&quot;:&quot;changes&quot;,&quot;scope&quot;:&quot;knockout-component-fbef08bd-21d6-4ec3-b3b9-28989f3a311d&quot;,&quot;id&quot;:&quot;ob_41&quot;,&quot;type&quot;:&quot;observable&quot;},val)) : undefined; return this.valueFromJulia[&quot;changes&quot;]=false}),self),this[&quot;value&quot;].subscribe((function (val){!(this.valueFromJulia[&quot;value&quot;]) ? (WebIO.setval({&quot;name&quot;:&quot;value&quot;,&quot;scope&quot;:&quot;knockout-component-fbef08bd-21d6-4ec3-b3b9-28989f3a311d&quot;,&quot;id&quot;:&quot;ob_40&quot;,&quot;type&quot;:&quot;observable&quot;},val)) : undefined; return this.valueFromJulia[&quot;value&quot;]=false}),self)]\n",
       "        \n",
       "    }\n",
       "    self.model = new AppViewModel();\n",
       "    self.valueFromJulia = {};\n",
       "    for (var key in json_data) {\n",
       "        self.valueFromJulia[key] = false;\n",
       "    }\n",
       "    ko.applyBindings(self.model, self.dom);\n",
       "}\n",
       "]},&quot;changes&quot;:[(function (val){return (val!=this.model[&quot;changes&quot;]()) ? (this.valueFromJulia[&quot;changes&quot;]=true, this.model[&quot;changes&quot;](val)) : undefined})],&quot;value&quot;:[(function (val){return (val!=this.model[&quot;value&quot;]()) ? (this.valueFromJulia[&quot;value&quot;]=true, this.model[&quot;value&quot;](val)) : undefined})]},&quot;systemjs_options&quot;:null,&quot;observables&quot;:{&quot;changes&quot;:{&quot;sync&quot;:false,&quot;id&quot;:&quot;ob_41&quot;,&quot;value&quot;:0},&quot;value&quot;:{&quot;sync&quot;:true,&quot;id&quot;:&quot;ob_40&quot;,&quot;value&quot;:14}}},&quot;children&quot;:[{&quot;props&quot;:{&quot;attributes&quot;:{&quot;class&quot;:&quot;interact-flex-row&quot;}},&quot;nodeType&quot;:&quot;DOM&quot;,&quot;type&quot;:&quot;node&quot;,&quot;instanceArgs&quot;:{&quot;namespace&quot;:&quot;html&quot;,&quot;tag&quot;:&quot;div&quot;},&quot;children&quot;:[{&quot;props&quot;:{&quot;attributes&quot;:{&quot;class&quot;:&quot;interact-flex-row-left&quot;}},&quot;nodeType&quot;:&quot;DOM&quot;,&quot;type&quot;:&quot;node&quot;,&quot;instanceArgs&quot;:{&quot;namespace&quot;:&quot;html&quot;,&quot;tag&quot;:&quot;div&quot;},&quot;children&quot;:[{&quot;props&quot;:{&quot;className&quot;:&quot;interact &quot;,&quot;style&quot;:{&quot;padding&quot;:&quot;5px 10px 0px 10px&quot;}},&quot;nodeType&quot;:&quot;DOM&quot;,&quot;type&quot;:&quot;node&quot;,&quot;instanceArgs&quot;:{&quot;namespace&quot;:&quot;html&quot;,&quot;tag&quot;:&quot;label&quot;},&quot;children&quot;:[&quot;j&quot;]}]},{&quot;props&quot;:{&quot;attributes&quot;:{&quot;class&quot;:&quot;interact-flex-row-center&quot;}},&quot;nodeType&quot;:&quot;DOM&quot;,&quot;type&quot;:&quot;node&quot;,&quot;instanceArgs&quot;:{&quot;namespace&quot;:&quot;html&quot;,&quot;tag&quot;:&quot;div&quot;},&quot;children&quot;:[{&quot;props&quot;:{&quot;max&quot;:28,&quot;min&quot;:1,&quot;attributes&quot;:{&quot;type&quot;:&quot;range&quot;,&quot;data-bind&quot;:&quot;numericValue: value, valueUpdate: &#39;input&#39;, event: {change : function () {this.changes(this.changes()+1)}}&quot;,&quot;orient&quot;:&quot;horizontal&quot;},&quot;step&quot;:1,&quot;className&quot;:&quot;slider slider is-fullwidth&quot;,&quot;style&quot;:{}},&quot;nodeType&quot;:&quot;DOM&quot;,&quot;type&quot;:&quot;node&quot;,&quot;instanceArgs&quot;:{&quot;namespace&quot;:&quot;html&quot;,&quot;tag&quot;:&quot;input&quot;},&quot;children&quot;:[]}]},{&quot;props&quot;:{&quot;attributes&quot;:{&quot;class&quot;:&quot;interact-flex-row-right&quot;}},&quot;nodeType&quot;:&quot;DOM&quot;,&quot;type&quot;:&quot;node&quot;,&quot;instanceArgs&quot;:{&quot;namespace&quot;:&quot;html&quot;,&quot;tag&quot;:&quot;div&quot;},&quot;children&quot;:[{&quot;props&quot;:{&quot;attributes&quot;:{&quot;data-bind&quot;:&quot;text: value&quot;}},&quot;nodeType&quot;:&quot;DOM&quot;,&quot;type&quot;:&quot;node&quot;,&quot;instanceArgs&quot;:{&quot;namespace&quot;:&quot;html&quot;,&quot;tag&quot;:&quot;p&quot;},&quot;children&quot;:[]}]}]}]}]},{&quot;props&quot;:{},&quot;nodeType&quot;:&quot;Scope&quot;,&quot;type&quot;:&quot;node&quot;,&quot;instanceArgs&quot;:{&quot;imports&quot;:{&quot;data&quot;:[],&quot;type&quot;:&quot;async_block&quot;},&quot;id&quot;:&quot;scope-22737198-45c5-47c6-9854-45b29fbc1658&quot;,&quot;handlers&quot;:{&quot;obs-output&quot;:[function (updated_htmlstr) {\n",
       "    var el = this.dom.querySelector(&quot;#out&quot;);\n",
       "    WebIO.propUtils.setInnerHtml(el, updated_htmlstr);\n",
       "}]},&quot;systemjs_options&quot;:null,&quot;observables&quot;:{&quot;obs-output&quot;:{&quot;sync&quot;:false,&quot;id&quot;:&quot;ob_44&quot;,&quot;value&quot;:&quot;&lt;div class=&#39;display:none&#39;&gt;&lt;/div&gt;&lt;unsafe-script style=&#39;display:none&#39;&gt;\\nWebIO.mount(this.previousSibling,{&amp;quot;props&amp;quot;:{&amp;quot;attributes&amp;quot;:{&amp;quot;class&amp;quot;:&amp;quot;interact-flex-row&amp;quot;}},&amp;quot;nodeType&amp;quot;:&amp;quot;DOM&amp;quot;,&amp;quot;type&amp;quot;:&amp;quot;node&amp;quot;,&amp;quot;instanceArgs&amp;quot;:{&amp;quot;namespace&amp;quot;:&amp;quot;html&amp;quot;,&amp;quot;tag&amp;quot;:&amp;quot;div&amp;quot;},&amp;quot;children&amp;quot;:[{&amp;quot;props&amp;quot;:{&amp;quot;style&amp;quot;:{&amp;quot;display&amp;quot;:&amp;quot;flex&amp;quot;,&amp;quot;flex-direction&amp;quot;:&amp;quot;row&amp;quot;}},&amp;quot;nodeType&amp;quot;:&amp;quot;DOM&amp;quot;,&amp;quot;type&amp;quot;:&amp;quot;node&amp;quot;,&amp;quot;instanceArgs&amp;quot;:{&amp;quot;namespace&amp;quot;:&amp;quot;html&amp;quot;,&amp;quot;tag&amp;quot;:&amp;quot;div&amp;quot;},&amp;quot;children&amp;quot;:[{&amp;quot;props&amp;quot;:{&amp;quot;style&amp;quot;:{&amp;quot;display&amp;quot;:&amp;quot;flex&amp;quot;,&amp;quot;flex-direction&amp;quot;:&amp;quot;column&amp;quot;}},&amp;quot;nodeType&amp;quot;:&amp;quot;DOM&amp;quot;,&amp;quot;type&amp;quot;:&amp;quot;node&amp;quot;,&amp;quot;instanceArgs&amp;quot;:{&amp;quot;namespace&amp;quot;:&amp;quot;html&amp;quot;,&amp;quot;tag&amp;quot;:&amp;quot;div&amp;quot;},&amp;quot;children&amp;quot;:[{&amp;quot;props&amp;quot;:{&amp;quot;setInnerHtml&amp;quot;:&amp;quot;&amp;lt;?xml version=\\\\&amp;quot;1.0\\\\&amp;quot; encoding=\\\\&amp;quot;UTF-8\\\\&amp;quot;?&amp;gt;\\\\n&amp;lt;!DOCTYPE svg PUBLIC \\\\&amp;quot;-//W3C//DTD SVG 1.1//EN\\\\&amp;quot;\\\\n \\\\&amp;quot;http://www.w3.org/Graphics/SVG/1.1/DTD/svg11.dtd\\\\&amp;quot;&amp;gt;\\\\n&amp;lt;svg xmlns=\\\\&amp;quot;http://www.w3.org/2000/svg\\\\&amp;quot; version=\\\\&amp;quot;1.1\\\\&amp;quot;\\\\n     width=\\\\&amp;quot;25mm\\\\&amp;quot; height=\\\\&amp;quot;25mm\\\\&amp;quot; viewBox=\\\\&amp;quot;0 0 1 1\\\\&amp;quot;&amp;gt;\\\\n     &amp;lt;rect width=\\\\&amp;quot;1\\\\&amp;quot; height=\\\\&amp;quot;1\\\\&amp;quot;\\\\n           fill=\\\\&amp;quot;#000000\\\\&amp;quot; stroke=\\\\&amp;quot;none\\\\&amp;quot;/&amp;gt;\\\\n&amp;lt;/svg&amp;gt;\\\\n&amp;quot;},&amp;quot;nodeType&amp;quot;:&amp;quot;DOM&amp;quot;,&amp;quot;type&amp;quot;:&amp;quot;node&amp;quot;,&amp;quot;instanceArgs&amp;quot;:{&amp;quot;namespace&amp;quot;:&amp;quot;html&amp;quot;,&amp;quot;tag&amp;quot;:&amp;quot;div&amp;quot;},&amp;quot;children&amp;quot;:[]},&amp;quot;0&amp;quot;]},{&amp;quot;props&amp;quot;:{&amp;quot;style&amp;quot;:{&amp;quot;display&amp;quot;:&amp;quot;flex&amp;quot;,&amp;quot;flex-direction&amp;quot;:&amp;quot;column&amp;quot;}},&amp;quot;nodeType&amp;quot;:&amp;quot;DOM&amp;quot;,&amp;quot;type&amp;quot;:&amp;quot;node&amp;quot;,&amp;quot;instanceArgs&amp;quot;:{&amp;quot;namespace&amp;quot;:&amp;quot;html&amp;quot;,&amp;quot;tag&amp;quot;:&amp;quot;div&amp;quot;},&amp;quot;children&amp;quot;:[{&amp;quot;props&amp;quot;:{&amp;quot;setInnerHtml&amp;quot;:&amp;quot;&amp;lt;?xml version=\\\\&amp;quot;1.0\\\\&amp;quot; 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       "14, nothing)), Set(String[]), nothing, Any[\"knockout\"=>\"/Users/alanedelman/.julia/packages/Knockout/JIqpG/src/../assets/knockout.js\", \"knockout_punches\"=>\"/Users/alanedelman/.julia/packages/Knockout/JIqpG/src/../assets/knockout_punches.js\", \"/Users/alanedelman/.julia/packages/InteractBase/PTCUD/src/../assets/all.js\", \"/Users/alanedelman/.julia/packages/InteractBase/PTCUD/src/../assets/style.css\", \"/Users/alanedelman/.julia/packages/InteractBulma/PH56C/src/../assets/main.css\"], Dict{Any,Any}(\"_promises\"=>Dict{Any,Any}(\"importsLoaded\"=>Any[JSString(\"function (ko, koPunches) {\\n    ko.punches.enableAll();\\n    ko.bindingHandlers.numericValue = {\\n        init : function(element, valueAccessor, allBindings, data, context) {\\n            var stringified = ko.observable(ko.unwrap(valueAccessor()));\\n            stringified.subscribe(function(value) {\\n                var val = parseFloat(value);\\n                if (!isNaN(val)) {\\n                    valueAccessor()(val);\\n                }\\n            })\\n            valueAccessor().subscribe(function(value) {\\n                var str = JSON.stringify(value);\\n                if ((str == \\\"0\\\") && ([\\\"-0\\\", \\\"-0.\\\"].indexOf(stringified()) >= 0))\\n                     return;\\n                 if ([\\\"null\\\", \\\"\\\"].indexOf(str) >= 0)\\n                     return;\\n                stringified(str);\\n            })\\n            ko.applyBindingsToNode(element, { value: stringified, valueUpdate: allBindings.get('valueUpdate')}, context);\\n        }\\n    };\\n    var json_data = JSON.parse(\\\"{\\\\\\\"changes\\\\\\\":0,\\\\\\\"value\\\\\\\":14}\\\");\\n    var self = this;\\n    function AppViewModel() {\\n        for (var key in json_data) {\\n            var el = json_data[key];\\n            this[key] = Array.isArray(el) ? ko.observableArray(el) : ko.observable(el);\\n        }\\n        \\n        \\n        [this[\\\"changes\\\"].subscribe((function (val){!(this.valueFromJulia[\\\"changes\\\"]) ? (WebIO.setval({\\\"name\\\":\\\"changes\\\",\\\"scope\\\":\\\"knockout-component-fbef08bd-21d6-4ec3-b3b9-28989f3a311d\\\",\\\"id\\\":\\\"ob_41\\\",\\\"type\\\":\\\"observable\\\"},val)) : undefined; return this.valueFromJulia[\\\"changes\\\"]=false}),self),this[\\\"value\\\"].subscribe((function (val){!(this.valueFromJulia[\\\"value\\\"]) ? (WebIO.setval({\\\"name\\\":\\\"value\\\",\\\"scope\\\":\\\"knockout-component-fbef08bd-21d6-4ec3-b3b9-28989f3a311d\\\",\\\"id\\\":\\\"ob_40\\\",\\\"type\\\":\\\"observable\\\"},val)) : undefined; return this.valueFromJulia[\\\"value\\\"]=false}),self)]\\n        \\n    }\\n    self.model = new AppViewModel();\\n    self.valueFromJulia = {};\\n    for (var key in json_data) {\\n        self.valueFromJulia[key] = false;\\n    }\\n    ko.applyBindings(self.model, self.dom);\\n}\\n\")]),\"changes\"=>Any[JSString(\"(function (val){return (val!=this.model[\\\"changes\\\"]()) ? (this.valueFromJulia[\\\"changes\\\"]=true, this.model[\\\"changes\\\"](val)) : undefined})\")],\"value\"=>Any[JSString(\"(function (val){return (val!=this.model[\\\"value\\\"]()) ? (this.valueFromJulia[\\\"value\\\"]=true, this.model[\\\"value\\\"](val)) : undefined})\")]), ConnectionPool(Channel{Any}(sz_max:9223372036854775807,sz_curr:3), Set(AbstractConnection[]), Channel{AbstractConnection}(sz_max:32,sz_curr:0))), ##52#53{#dom#15{##dom#13#14{Dict{Any,Any},DOM}},typeof(scope)}(#dom#15{##dom#13#14{Dict{Any,Any},DOM}}(##dom#13#14{Dict{Any,Any},DOM}(Dict{Any,Any}(:className=>\"field\"), DOM(:html, :div))), scope))), Observable{Any} with 0 listeners. Value:\n",
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svg PUBLIC \\\"-//W3C//DTD SVG 1.1//EN\\\"\\n \\\"http://www.w3.org/Graphics/SVG/1.1/DTD/svg11.dtd\\\">\\n<svg xmlns=\\\"http://www.w3.org/2000/svg\\\" version=\\\"1.1\\\"\\n     width=\\\"25mm\\\" height=\\\"25mm\\\" viewBox=\\\"0 0 1 1\\\">\\n     <rect width=\\\"1\\\" height=\\\"1\\\"\\n           fill=\\\"#000000\\\" stroke=\\\"none\\\"/>\\n</svg>\\n\"), 0), \"0\"], Dict{Symbol,Any}(:style=>Dict(\"display\"=>\"flex\",\"flex-direction\"=>\"column\")), 2), Node{DOM}(DOM(:html, :div), Any[Node{DOM}(DOM(:html, :div), Any[], Dict{Symbol,Any}(:setInnerHtml=>\"<?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?>\\n<!DOCTYPE svg PUBLIC \\\"-//W3C//DTD SVG 1.1//EN\\\"\\n \\\"http://www.w3.org/Graphics/SVG/1.1/DTD/svg11.dtd\\\">\\n<svg xmlns=\\\"http://www.w3.org/2000/svg\\\" version=\\\"1.1\\\"\\n     width=\\\"25mm\\\" height=\\\"25mm\\\" viewBox=\\\"0 0 1 1\\\">\\n     <rect width=\\\"1\\\" height=\\\"1\\\"\\n           fill=\\\"#393939\\\" stroke=\\\"none\\\"/>\\n</svg>\\n\"), 0), \"57\"], Dict{Symbol,Any}(:style=>Dict(\"display\"=>\"flex\",\"flex-direction\"=>\"column\")), 2), Node{DOM}(DOM(:html, :div), Any[Node{DOM}(DOM(:html, :div), Any[], Dict{Symbol,Any}(:setInnerHtml=>\"<?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?>\\n<!DOCTYPE svg PUBLIC \\\"-//W3C//DTD SVG 1.1//EN\\\"\\n \\\"http://www.w3.org/Graphics/SVG/1.1/DTD/svg11.dtd\\\">\\n<svg xmlns=\\\"http://www.w3.org/2000/svg\\\" version=\\\"1.1\\\"\\n     width=\\\"25mm\\\" height=\\\"25mm\\\" viewBox=\\\"0 0 1 1\\\">\\n     <rect width=\\\"1\\\" height=\\\"1\\\"\\n           fill=\\\"#FCFCFC\\\" stroke=\\\"none\\\"/>\\n</svg>\\n\"), 0), \"253\"], Dict{Symbol,Any}(:style=>Dict(\"display\"=>\"flex\",\"flex-direction\"=>\"column\")), 2), Node{DOM}(DOM(:html, :div), Any[Node{DOM}(DOM(:html, :div), Any[], Dict{Symbol,Any}(:setInnerHtml=>\"<?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?>\\n<!DOCTYPE svg PUBLIC \\\"-//W3C//DTD SVG 1.1//EN\\\"\\n \\\"http://www.w3.org/Graphics/SVG/1.1/DTD/svg11.dtd\\\">\\n<svg xmlns=\\\"http://www.w3.org/2000/svg\\\" version=\\\"1.1\\\"\\n     width=\\\"25mm\\\" height=\\\"25mm\\\" viewBox=\\\"0 0 1 1\\\">\\n     <rect width=\\\"1\\\" height=\\\"1\\\"\\n           fill=\\\"#FCFCFC\\\" stroke=\\\"none\\\"/>\\n</svg>\\n\"), 0), \"253\"], Dict{Symbol,Any}(:style=>Dict(\"display\"=>\"flex\",\"flex-direction\"=>\"column\")), 2)  …  Node{DOM}(DOM(:html, :div), Any[Node{DOM}(DOM(:html, :div), Any[], Dict{Symbol,Any}(:setInnerHtml=>\"<?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?>\\n<!DOCTYPE svg PUBLIC \\\"-//W3C//DTD SVG 1.1//EN\\\"\\n \\\"http://www.w3.org/Graphics/SVG/1.1/DTD/svg11.dtd\\\">\\n<svg xmlns=\\\"http://www.w3.org/2000/svg\\\" version=\\\"1.1\\\"\\n     width=\\\"25mm\\\" height=\\\"25mm\\\" viewBox=\\\"0 0 1 1\\\">\\n     <rect width=\\\"1\\\" height=\\\"1\\\"\\n           fill=\\\"#000000\\\" stroke=\\\"none\\\"/>\\n</svg>\\n\"), 0), \"0\"], Dict{Symbol,Any}(:style=>Dict(\"display\"=>\"flex\",\"flex-direction\"=>\"column\")), 2), Node{DOM}(DOM(:html, :div), Any[Node{DOM}(DOM(:html, :div), Any[], Dict{Symbol,Any}(:setInnerHtml=>\"<?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?>\\n<!DOCTYPE svg PUBLIC \\\"-//W3C//DTD SVG 1.1//EN\\\"\\n \\\"http://www.w3.org/Graphics/SVG/1.1/DTD/svg11.dtd\\\">\\n<svg xmlns=\\\"http://www.w3.org/2000/svg\\\" version=\\\"1.1\\\"\\n     width=\\\"25mm\\\" height=\\\"25mm\\\" viewBox=\\\"0 0 1 1\\\">\\n     <rect width=\\\"1\\\" height=\\\"1\\\"\\n           fill=\\\"#000000\\\" stroke=\\\"none\\\"/>\\n</svg>\\n\"), 0), \"0\"], Dict{Symbol,Any}(:style=>Dict(\"display\"=>\"flex\",\"flex-direction\"=>\"column\")), 2), Node{DOM}(DOM(:html, :div), Any[Node{DOM}(DOM(:html, :div), Any[], Dict{Symbol,Any}(:setInnerHtml=>\"<?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?>\\n<!DOCTYPE svg PUBLIC \\\"-//W3C//DTD SVG 1.1//EN\\\"\\n \\\"http://www.w3.org/Graphics/SVG/1.1/DTD/svg11.dtd\\\">\\n<svg xmlns=\\\"http://www.w3.org/2000/svg\\\" version=\\\"1.1\\\"\\n     width=\\\"25mm\\\" height=\\\"25mm\\\" viewBox=\\\"0 0 1 1\\\">\\n     <rect width=\\\"1\\\" height=\\\"1\\\"\\n           fill=\\\"#FDFDFD\\\" stroke=\\\"none\\\"/>\\n</svg>\\n\"), 0), \"254\"], Dict{Symbol,Any}(:style=>Dict(\"display\"=>\"flex\",\"flex-direction\"=>\"column\")), 2), Node{DOM}(DOM(:html, :div), Any[Node{DOM}(DOM(:html, :div), Any[], Dict{Symbol,Any}(:setInnerHtml=>\"<?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?>\\n<!DOCTYPE svg PUBLIC \\\"-//W3C//DTD SVG 1.1//EN\\\"\\n \\\"http://www.w3.org/Graphics/SVG/1.1/DTD/svg11.dtd\\\">\\n<svg xmlns=\\\"http://www.w3.org/2000/svg\\\" version=\\\"1.1\\\"\\n     width=\\\"25mm\\\" height=\\\"25mm\\\" viewBox=\\\"0 0 1 1\\\">\\n     <rect width=\\\"1\\\" height=\\\"1\\\"\\n           fill=\\\"#FCFCFC\\\" stroke=\\\"none\\\"/>\\n</svg>\\n\"), 0), \"253\"], Dict{Symbol,Any}(:style=>Dict(\"display\"=>\"flex\",\"flex-direction\"=>\"column\")), 2), Node{DOM}(DOM(:html, :div), Any[Node{DOM}(DOM(:html, :div), Any[], Dict{Symbol,Any}(:setInnerHtml=>\"<?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?>\\n<!DOCTYPE svg PUBLIC \\\"-//W3C//DTD SVG 1.1//EN\\\"\\n \\\"http://www.w3.org/Graphics/SVG/1.1/DTD/svg11.dtd\\\">\\n<svg xmlns=\\\"http://www.w3.org/2000/svg\\\" version=\\\"1.1\\\"\\n     width=\\\"25mm\\\" height=\\\"25mm\\\" viewBox=\\\"0 0 1 1\\\">\\n     <rect width=\\\"1\\\" height=\\\"1\\\"\\n           fill=\\\"#C3C3C3\\\" stroke=\\\"none\\\"/>\\n</svg>\\n\"), 0), \"196\"], Dict{Symbol,Any}(:style=>Dict(\"display\"=>\"flex\",\"flex-direction\"=>\"column\")), 2), Node{DOM}(DOM(:html, :div), Any[Node{DOM}(DOM(:html, :div), Any[], Dict{Symbol,Any}(:setInnerHtml=>\"<?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?>\\n<!DOCTYPE svg PUBLIC \\\"-//W3C//DTD SVG 1.1//EN\\\"\\n \\\"http://www.w3.org/Graphics/SVG/1.1/DTD/svg11.dtd\\\">\\n<svg xmlns=\\\"http://www.w3.org/2000/svg\\\" version=\\\"1.1\\\"\\n     width=\\\"25mm\\\" height=\\\"25mm\\\" viewBox=\\\"0 0 1 1\\\">\\n     <rect width=\\\"1\\\" height=\\\"1\\\"\\n           fill=\\\"#000000\\\" stroke=\\\"none\\\"/>\\n</svg>\\n\"), 0), \"0\"], Dict{Symbol,Any}(:style=>Dict(\"display\"=>\"flex\",\"flex-direction\"=>\"column\")), 2), Node{DOM}(DOM(:html, :div), Any[Node{DOM}(DOM(:html, :div), Any[], Dict{Symbol,Any}(:setInnerHtml=>\"<?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?>\\n<!DOCTYPE svg PUBLIC \\\"-//W3C//DTD SVG 1.1//EN\\\"\\n \\\"http://www.w3.org/Graphics/SVG/1.1/DTD/svg11.dtd\\\">\\n<svg xmlns=\\\"http://www.w3.org/2000/svg\\\" version=\\\"1.1\\\"\\n     width=\\\"25mm\\\" height=\\\"25mm\\\" viewBox=\\\"0 0 1 1\\\">\\n     <rect width=\\\"1\\\" height=\\\"1\\\"\\n           fill=\\\"#000000\\\" stroke=\\\"none\\\"/>\\n</svg>\\n\"), 0), \"0\"], Dict{Symbol,Any}(:style=>Dict(\"display\"=>\"flex\",\"flex-direction\"=>\"column\")), 2), Node{DOM}(DOM(:html, :div), Any[Node{DOM}(DOM(:html, :div), Any[], Dict{Symbol,Any}(:setInnerHtml=>\"<?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?>\\n<!DOCTYPE svg PUBLIC \\\"-//W3C//DTD SVG 1.1//EN\\\"\\n \\\"http://www.w3.org/Graphics/SVG/1.1/DTD/svg11.dtd\\\">\\n<svg xmlns=\\\"http://www.w3.org/2000/svg\\\" version=\\\"1.1\\\"\\n     width=\\\"25mm\\\" height=\\\"25mm\\\" viewBox=\\\"0 0 1 1\\\">\\n     <rect width=\\\"1\\\" height=\\\"1\\\"\\n           fill=\\\"#000000\\\" stroke=\\\"none\\\"/>\\n</svg>\\n\"), 0), \"0\"], Dict{Symbol,Any}(:style=>Dict(\"display\"=>\"flex\",\"flex-direction\"=>\"column\")), 2), Node{DOM}(DOM(:html, :div), Any[Node{DOM}(DOM(:html, :div), Any[], Dict{Symbol,Any}(:setInnerHtml=>\"<?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?>\\n<!DOCTYPE svg PUBLIC \\\"-//W3C//DTD SVG 1.1//EN\\\"\\n \\\"http://www.w3.org/Graphics/SVG/1.1/DTD/svg11.dtd\\\">\\n<svg xmlns=\\\"http://www.w3.org/2000/svg\\\" version=\\\"1.1\\\"\\n     width=\\\"25mm\\\" height=\\\"25mm\\\" viewBox=\\\"0 0 1 1\\\">\\n     <rect width=\\\"1\\\" height=\\\"1\\\"\\n           fill=\\\"#000000\\\" stroke=\\\"none\\\"/>\\n</svg>\\n\"), 0), \"0\"], Dict{Symbol,Any}(:style=>Dict(\"display\"=>\"flex\",\"flex-direction\"=>\"column\")), 2), Node{DOM}(DOM(:html, :div), Any[Node{DOM}(DOM(:html, :div), Any[], Dict{Symbol,Any}(:setInnerHtml=>\"<?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?>\\n<!DOCTYPE svg PUBLIC \\\"-//W3C//DTD SVG 1.1//EN\\\"\\n \\\"http://www.w3.org/Graphics/SVG/1.1/DTD/svg11.dtd\\\">\\n<svg xmlns=\\\"http://www.w3.org/2000/svg\\\" version=\\\"1.1\\\"\\n     width=\\\"25mm\\\" height=\\\"25mm\\\" viewBox=\\\"0 0 1 1\\\">\\n     <rect width=\\\"1\\\" height=\\\"1\\\"\\n           fill=\\\"#000000\\\" stroke=\\\"none\\\"/>\\n</svg>\\n\"), 0), \"0\"], Dict{Symbol,Any}(:style=>Dict(\"display\"=>\"flex\",\"flex-direction\"=>\"column\")), 2)], Dict{Symbol,Any}(:style=>Dict(\"display\"=>\"flex\",\"flex-direction\"=>\"row\")), 84), nothing, getfield(InteractBase, Symbol(\"##142#143\"))())"
      ]
     },
     "execution_count": 48,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "@manipulate for j in 1:28\n",
    "    a = vec(images[2][j, :])\n",
    "    b = round.(Int, 256*images[2][j, i].val for i in 1:28);\n",
    "    \n",
    "    hbox([vbox(a[i], string(b[i])) for i in 1:28])\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 49,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<div class='tex2jax_ignore interactbulma'>\n",
       "<div class='display:none'></div><unsafe-script style='display:none'>\n",
       "WebIO.mount(this.previousSibling,{&quot;props&quot;:{},&quot;nodeType&quot;:&quot;DOM&quot;,&quot;type&quot;:&quot;node&quot;,&quot;instanceArgs&quot;:{&quot;namespace&quot;:&quot;html&quot;,&quot;tag&quot;:&quot;div&quot;},&quot;children&quot;:[{&quot;props&quot;:{&quot;className&quot;:&quot;field&quot;},&quot;nodeType&quot;:&quot;DOM&quot;,&quot;type&quot;:&quot;node&quot;,&quot;instanceArgs&quot;:{&quot;namespace&quot;:&quot;html&quot;,&quot;tag&quot;:&quot;div&quot;},&quot;children&quot;:[{&quot;props&quot;:{},&quot;nodeType&quot;:&quot;Scope&quot;,&quot;type&quot;:&quot;node&quot;,&quot;instanceArgs&quot;:{&quot;imports&quot;:{&quot;data&quot;:[{&quot;name&quot;:&quot;knockout&quot;,&quot;type&quot;:&quot;js&quot;,&quot;url&quot;:&quot;/assetserver/06c67466ebc6c52906ffd4101b7c74a47f9567bc-knockout.js&quot;},{&quot;name&quot;:&quot;knockout_punches&quot;,&quot;type&quot;:&quot;js&quot;,&quot;url&quot;:&quot;/assetserver/46b0ee2b2af2730ca29ddb46f091ac108fb8f609-knockout_punches.js&quot;},{&quot;name&quot;:null,&quot;type&quot;:&quot;js&quot;,&quot;url&quot;:&quot;/assetserver/e923a2cf7cc8bb183f572d980ae606c5f08fa0dc-all.js&quot;},{&quot;name&quot;:null,&quot;type&quot;:&quot;css&quot;,&quot;url&quot;:&quot;/assetserver/b973c333a7d1e5a6ca74f1442a2f26b5e4d0580e-style.css&quot;},{&quot;name&quot;:null,&quot;type&quot;:&quot;css&quot;,&quot;url&quot;:&quot;/assetserver/885f9c98e35e2ecc817c654a867b4570a606ec22-main.css&quot;}],&quot;type&quot;:&quot;async_block&quot;},&quot;id&quot;:&quot;knockout-component-e4af6641-2c17-46a5-bccd-0ba6067c8532&quot;,&quot;handlers&quot;:{&quot;_promises&quot;:{&quot;importsLoaded&quot;:[function (ko, koPunches) {\n",
       "    ko.punches.enableAll();\n",
       "    ko.bindingHandlers.numericValue = {\n",
       "        init : function(element, valueAccessor, allBindings, data, context) {\n",
       "            var stringified = ko.observable(ko.unwrap(valueAccessor()));\n",
       "            stringified.subscribe(function(value) {\n",
       "                var val = parseFloat(value);\n",
       "                if (!isNaN(val)) {\n",
       "                    valueAccessor()(val);\n",
       "                }\n",
       "            })\n",
       "            valueAccessor().subscribe(function(value) {\n",
       "                var str = JSON.stringify(value);\n",
       "                if ((str == &quot;0&quot;) &amp;&amp; ([&quot;-0&quot;, &quot;-0.&quot;].indexOf(stringified()) &gt;= 0))\n",
       "                     return;\n",
       "                 if ([&quot;null&quot;, &quot;&quot;].indexOf(str) &gt;= 0)\n",
       "                     return;\n",
       "                stringified(str);\n",
       "            })\n",
       "            ko.applyBindingsToNode(element, { value: stringified, valueUpdate: allBindings.get(&#39;valueUpdate&#39;)}, context);\n",
       "        }\n",
       "    };\n",
       "    var json_data = JSON.parse(&quot;{\\&quot;changes\\&quot;:0,\\&quot;value\\&quot;:378}&quot;);\n",
       "    var self = this;\n",
       "    function AppViewModel() {\n",
       "        for (var key in json_data) {\n",
       "            var el = json_data[key];\n",
       "            this[key] = Array.isArray(el) ? ko.observableArray(el) : ko.observable(el);\n",
       "        }\n",
       "        \n",
       "        \n",
       "        [this[&quot;changes&quot;].subscribe((function (val){!(this.valueFromJulia[&quot;changes&quot;]) ? (WebIO.setval({&quot;name&quot;:&quot;changes&quot;,&quot;scope&quot;:&quot;knockout-component-e4af6641-2c17-46a5-bccd-0ba6067c8532&quot;,&quot;id&quot;:&quot;ob_46&quot;,&quot;type&quot;:&quot;observable&quot;},val)) : undefined; return this.valueFromJulia[&quot;changes&quot;]=false}),self),this[&quot;value&quot;].subscribe((function (val){!(this.valueFromJulia[&quot;value&quot;]) ? (WebIO.setval({&quot;name&quot;:&quot;value&quot;,&quot;scope&quot;:&quot;knockout-component-e4af6641-2c17-46a5-bccd-0ba6067c8532&quot;,&quot;id&quot;:&quot;ob_45&quot;,&quot;type&quot;:&quot;observable&quot;},val)) : undefined; return this.valueFromJulia[&quot;value&quot;]=false}),self)]\n",
       "        \n",
       "    }\n",
       "    self.model = new AppViewModel();\n",
       "    self.valueFromJulia = {};\n",
       "    for (var key in json_data) {\n",
       "        self.valueFromJulia[key] = false;\n",
       "    }\n",
       "    ko.applyBindings(self.model, self.dom);\n",
       "}\n",
       "]},&quot;changes&quot;:[(function (val){return (val!=this.model[&quot;changes&quot;]()) ? (this.valueFromJulia[&quot;changes&quot;]=true, this.model[&quot;changes&quot;](val)) : undefined})],&quot;value&quot;:[(function (val){return (val!=this.model[&quot;value&quot;]()) ? (this.valueFromJulia[&quot;value&quot;]=true, this.model[&quot;value&quot;](val)) : undefined})]},&quot;systemjs_options&quot;:null,&quot;observables&quot;:{&quot;changes&quot;:{&quot;sync&quot;:false,&quot;id&quot;:&quot;ob_46&quot;,&quot;value&quot;:0},&quot;value&quot;:{&quot;sync&quot;:true,&quot;id&quot;:&quot;ob_45&quot;,&quot;value&quot;:378}}},&quot;children&quot;:[{&quot;props&quot;:{&quot;attributes&quot;:{&quot;class&quot;:&quot;interact-flex-row&quot;}},&quot;nodeType&quot;:&quot;DOM&quot;,&quot;type&quot;:&quot;node&quot;,&quot;instanceArgs&quot;:{&quot;namespace&quot;:&quot;html&quot;,&quot;tag&quot;:&quot;div&quot;},&quot;children&quot;:[{&quot;props&quot;:{&quot;attributes&quot;:{&quot;class&quot;:&quot;interact-flex-row-left&quot;}},&quot;nodeType&quot;:&quot;DOM&quot;,&quot;type&quot;:&quot;node&quot;,&quot;instanceArgs&quot;:{&quot;namespace&quot;:&quot;html&quot;,&quot;tag&quot;:&quot;div&quot;},&quot;children&quot;:[{&quot;props&quot;:{&quot;className&quot;:&quot;interact &quot;,&quot;style&quot;:{&quot;padding&quot;:&quot;5px 10px 0px 10px&quot;}},&quot;nodeType&quot;:&quot;DOM&quot;,&quot;type&quot;:&quot;node&quot;,&quot;instanceArgs&quot;:{&quot;namespace&quot;:&quot;html&quot;,&quot;tag&quot;:&quot;label&quot;},&quot;children&quot;:[&quot;start&quot;]}]},{&quot;props&quot;:{&quot;attributes&quot;:{&quot;class&quot;:&quot;interact-flex-row-center&quot;}},&quot;nodeType&quot;:&quot;DOM&quot;,&quot;type&quot;:&quot;node&quot;,&quot;instanceArgs&quot;:{&quot;namespace&quot;:&quot;html&quot;,&quot;tag&quot;:&quot;div&quot;},&quot;children&quot;:[{&quot;props&quot;:{&quot;max&quot;:756,&quot;min&quot;:1,&quot;attributes&quot;:{&quot;type&quot;:&quot;range&quot;,&quot;data-bind&quot;:&quot;numericValue: value, valueUpdate: &#39;input&#39;, event: {change : function () {this.changes(this.changes()+1)}}&quot;,&quot;orient&quot;:&quot;horizontal&quot;},&quot;step&quot;:1,&quot;className&quot;:&quot;slider slider is-fullwidth&quot;,&quot;style&quot;:{}},&quot;nodeType&quot;:&quot;DOM&quot;,&quot;type&quot;:&quot;node&quot;,&quot;instanceArgs&quot;:{&quot;namespace&quot;:&quot;html&quot;,&quot;tag&quot;:&quot;input&quot;},&quot;children&quot;:[]}]},{&quot;props&quot;:{&quot;attributes&quot;:{&quot;class&quot;:&quot;interact-flex-row-right&quot;}},&quot;nodeType&quot;:&quot;DOM&quot;,&quot;type&quot;:&quot;node&quot;,&quot;instanceArgs&quot;:{&quot;namespace&quot;:&quot;html&quot;,&quot;tag&quot;:&quot;div&quot;},&quot;children&quot;:[{&quot;props&quot;:{&quot;attributes&quot;:{&quot;data-bind&quot;:&quot;text: value&quot;}},&quot;nodeType&quot;:&quot;DOM&quot;,&quot;type&quot;:&quot;node&quot;,&quot;instanceArgs&quot;:{&quot;namespace&quot;:&quot;html&quot;,&quot;tag&quot;:&quot;p&quot;},&quot;children&quot;:[]}]}]}]}]},{&quot;props&quot;:{},&quot;nodeType&quot;:&quot;Scope&quot;,&quot;type&quot;:&quot;node&quot;,&quot;instanceArgs&quot;:{&quot;imports&quot;:{&quot;data&quot;:[],&quot;type&quot;:&quot;async_block&quot;},&quot;id&quot;:&quot;scope-91d9936d-9c91-41c4-b7d7-8dbdb8f634c1&quot;,&quot;handlers&quot;:{&quot;obs-output&quot;:[function (updated_htmlstr) {\n",
       "    var el = this.dom.querySelector(&quot;#out&quot;);\n",
       "    WebIO.propUtils.setInnerHtml(el, updated_htmlstr);\n",
       "}]},&quot;systemjs_options&quot;:null,&quot;observables&quot;:{&quot;obs-output&quot;:{&quot;sync&quot;:false,&quot;id&quot;:&quot;ob_49&quot;,&quot;value&quot;:&quot;&lt;div class=&#39;display:none&#39;&gt;&lt;/div&gt;&lt;unsafe-script style=&#39;display:none&#39;&gt;\\nWebIO.mount(this.previousSibling,{&amp;quot;props&amp;quot;:{&amp;quot;attributes&amp;quot;:{&amp;quot;class&amp;quot;:&amp;quot;interact-flex-row&amp;quot;}},&amp;quot;nodeType&amp;quot;:&amp;quot;DOM&amp;quot;,&amp;quot;type&amp;quot;:&amp;quot;node&amp;quot;,&amp;quot;instanceArgs&amp;quot;:{&amp;quot;namespace&amp;quot;:&amp;quot;html&amp;quot;,&amp;quot;tag&amp;quot;:&amp;quot;div&amp;quot;},&amp;quot;children&amp;quot;:[{&amp;quot;props&amp;quot;:{&amp;quot;style&amp;quot;:{&amp;quot;display&amp;quot;:&amp;quot;flex&amp;quot;,&amp;quot;flex-direction&amp;quot;:&amp;quot;row&amp;quot;}},&amp;quot;nodeType&amp;quot;:&amp;quot;DOM&amp;quot;,&amp;quot;type&amp;quot;:&amp;quot;node&amp;quot;,&amp;quot;instanceArgs&amp;quot;:{&amp;quot;namespace&amp;quot;:&amp;quot;html&amp;quot;,&amp;quot;tag&amp;quot;:&amp;quot;div&amp;quot;},&amp;quot;children&amp;quot;:[{&amp;quot;props&amp;quot;:{&amp;quot;style&amp;quot;:{&amp;quot;display&amp;quot;:&amp;quot;flex&amp;quot;,&amp;quot;flex-direction&amp;quot;:&amp;quot;column&amp;quot;}},&amp;quot;nodeType&amp;quot;:&amp;quot;DOM&amp;quot;,&amp;quot;type&amp;quot;:&amp;quot;node&amp;quot;,&amp;quot;instanceArgs&amp;quot;:{&amp;quot;namespace&amp;quot;:&amp;quot;html&amp;quot;,&amp;quot;tag&amp;quot;:&amp;quot;div&amp;quot;},&amp;quot;children&amp;quot;:[{&amp;quot;props&amp;quot;:{&amp;quot;setInnerHtml&amp;quot;:&amp;quot;&amp;lt;?xml version=\\\\&amp;quot;1.0\\\\&amp;quot; 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      ]
     },
     "execution_count": 49,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "@manipulate for start in 1:28^2 - 28\n",
    "    a = vec(images[2][start:start+27])\n",
    "    b = round.(Int, 256*images[2][i].val for i in start:start+27);\n",
    "    \n",
    "    hbox([vbox(a[i], hbox(hskip(0.5em), string(b[i]))) for i in 1:28])\n",
    "end"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Collect all the data into a matrix"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's collect all the data into a matrix. We take 50000 images as our training set:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 116,
   "metadata": {},
   "outputs": [],
   "source": [
    "training_size = 50000\n",
    "\n",
    "training_images = images[1:training_size]\n",
    "training_labels = labels[1:training_size];"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "For example, let's look at the images of 0s:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 117,
   "metadata": {},
   "outputs": [
    {
     "data": {
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max-height:100px;display:inline' src=\"data:image/png;base64,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\"></td><td style='text-align:center;vertical-align:middle; 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margin: 0.5em;border:1px #90999f solid;border-collapse:collapse'><img style='max-width: 100px; max-height:100px;display:inline' src=\"data:image/png;base64,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\"></td><td style='text-align:center;vertical-align:middle; margin: 0.5em;border:1px #90999f solid;border-collapse:collapse'><img style='max-width: 100px; max-height:100px;display:inline' src=\"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAHAAAABwCAAAAADji6uXAAAABGdBTUEAALGPC/xhBQAAACBjSFJNAAB6JgAAgIQAAPoAAACA6AAAdTAAAOpgAAA6mAAAF3CculE8AAAAAmJLR0QA/4ePzL8AAAJBSURBVGje7dpNiE5hGAbgi5lRoi81s5TNoBlFSlYWyuyULJT8pclK2JiaUhYoLBQLYrJSSikpq1EWjJoFJWUzaSK/uymU/IxhsniP5vQ53/GZxXxn3t579TznfU73ubt7zvtzDgkJCQkJCQkLZntjOzbhPDbgPW7jCsZL7ls41wrjJ5yVh904g50FY59xAaeqojB+wll5eAQXS8Y/oBcTVVAYP2H7/xR3YAj9uWtfcVN4lx7DInxHp+ThHKFpD9cJ8113lk9iAHfxGvtytZ1YiedVUBg/YdMebjPjH2HOG8rlI/gm9OEw7ldFYfyETXl4B1tz+TOcrKsZQC2LV+BXVRTGT/hPD5djS65wHLsxVVf3yswCqQuLhfdtyxXGT1jqYa8wBy7N8qc46O95rh2bc/lVfKqKwvgJSz28jp4sfoNBPGnw1GuzeBoPq6QwfsKGHvZgfS6/hwcNandhVRZPK16Ptkxh/IQNPRxEWxaP4lCDuhqO5/K3+FglhfETFnq4DH1ZPCWsQYvWmTWcM9ODkzhRNYXxExZ6uEfYHxB6sGiv1yHsEQ/krr3AjaopjJ+w0MOuXPyoYLwmrFf7ctcmsL+KCuMnLPQwfx7zMhe3CX13VjhP+4NRbFc+D7ZMYfyEhR4OY2/2NKuF9+ZR7MDGXN1PXMYlzfnXEoXxExZ+e1qCx1hTcuM1Ya3zruoK4ycs7MMvuCX0Xq1ubEw4Gx3Bj/mgMH7C0m/A/Tgs/GsBp4VvhWPzSWH8hAkJCQkJCQn8BtasVgERhAluAAAAAElFTkSuQmCC\"></td><td style='text-align:center;vertical-align:middle; margin: 0.5em;border:1px #90999f solid;border-collapse:collapse'><img style='max-width: 100px; max-height:100px;display:inline' src=\"data:image/png;base64,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\"></td><td style='text-align:center;vertical-align:middle; margin: 0.5em;border:1px #90999f solid;border-collapse:collapse'><img style='max-width: 100px; max-height:100px;display:inline' src=\"data:image/png;base64,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\"></td><td style='text-align:center;vertical-align:middle; margin: 0.5em;border:1px #90999f solid;border-collapse:collapse'><img style='max-width: 100px; max-height:100px;display:inline' src=\"data:image/png;base64,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\"></td><td style='text-align:center;vertical-align:middle; margin: 0.5em;border:1px #90999f solid;border-collapse:collapse'><img style='max-width: 100px; max-height:100px;display:inline' src=\"data:image/png;base64,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\"></td><td style='text-align:center;vertical-align:middle; margin: 0.5em;border:1px #90999f solid;border-collapse:collapse'><img style='max-width: 100px; max-height:100px;display:inline' src=\"data:image/png;base64,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\"></td><td style='text-align:center;vertical-align:middle; margin: 0.5em;border:1px #90999f solid;border-collapse:collapse'><img style='max-width: 100px; max-height:100px;display:inline' src=\"data:image/png;base64,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\"></td><td style='text-align:center;vertical-align:middle; margin: 0.5em;border:1px #90999f solid;border-collapse:collapse'><img style='max-width: 100px; max-height:100px;display:inline' src=\"data:image/png;base64,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\"></td><td style='text-align:center;vertical-align:middle; margin: 0.5em;border:1px #90999f solid;border-collapse:collapse'><img style='max-width: 100px; max-height:100px;display:inline' src=\"data:image/png;base64,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\"></td><td style='text-align:center;vertical-align:middle; margin: 0.5em;border:1px #90999f solid;border-collapse:collapse'><img style='max-width: 100px; max-height:100px;display:inline' src=\"data:image/png;base64,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\"></td></tr></tbody></table><div><small>(a vector displayed as a row to save space)</small></div>"
      ],
      "text/plain": [
       "100-element Array{Array{Gray{FixedPointNumbers.Normed{UInt8,8}},2},1}:\n",
       " [Gray{N0f8}(0.0) Gray{N0f8}(0.0) … Gray{N0f8}(0.0) Gray{N0f8}(0.0); Gray{N0f8}(0.0) Gray{N0f8}(0.0) … Gray{N0f8}(0.0) Gray{N0f8}(0.0); … ; Gray{N0f8}(0.0) Gray{N0f8}(0.0) … Gray{N0f8}(0.0) Gray{N0f8}(0.0); Gray{N0f8}(0.0) Gray{N0f8}(0.0) … Gray{N0f8}(0.0) Gray{N0f8}(0.0)]\n",
       " [Gray{N0f8}(0.0) Gray{N0f8}(0.0) … Gray{N0f8}(0.0) Gray{N0f8}(0.0); Gray{N0f8}(0.0) Gray{N0f8}(0.0) … Gray{N0f8}(0.0) Gray{N0f8}(0.0); … ; Gray{N0f8}(0.0) Gray{N0f8}(0.0) … Gray{N0f8}(0.0) Gray{N0f8}(0.0); Gray{N0f8}(0.0) Gray{N0f8}(0.0) … Gray{N0f8}(0.0) Gray{N0f8}(0.0)]\n",
       " [Gray{N0f8}(0.0) Gray{N0f8}(0.0) … Gray{N0f8}(0.0) Gray{N0f8}(0.0); Gray{N0f8}(0.0) Gray{N0f8}(0.0) … Gray{N0f8}(0.0) Gray{N0f8}(0.0); … ; Gray{N0f8}(0.0) Gray{N0f8}(0.0) … Gray{N0f8}(0.0) Gray{N0f8}(0.0); Gray{N0f8}(0.0) Gray{N0f8}(0.0) … Gray{N0f8}(0.0) Gray{N0f8}(0.0)]\n",
       " [Gray{N0f8}(0.0) Gray{N0f8}(0.0) … Gray{N0f8}(0.0) Gray{N0f8}(0.0); Gray{N0f8}(0.0) Gray{N0f8}(0.0) … Gray{N0f8}(0.0) Gray{N0f8}(0.0); … ; Gray{N0f8}(0.0) Gray{N0f8}(0.0) … Gray{N0f8}(0.0) Gray{N0f8}(0.0); Gray{N0f8}(0.0) Gray{N0f8}(0.0) … Gray{N0f8}(0.0) Gray{N0f8}(0.0)]\n",
       " [Gray{N0f8}(0.0) Gray{N0f8}(0.0) … Gray{N0f8}(0.0) Gray{N0f8}(0.0); Gray{N0f8}(0.0) Gray{N0f8}(0.0) … Gray{N0f8}(0.0) Gray{N0f8}(0.0); … ; Gray{N0f8}(0.0) Gray{N0f8}(0.0) … Gray{N0f8}(0.0) Gray{N0f8}(0.0); Gray{N0f8}(0.0) Gray{N0f8}(0.0) … Gray{N0f8}(0.0) Gray{N0f8}(0.0)]\n",
       " [Gray{N0f8}(0.0) Gray{N0f8}(0.0) … Gray{N0f8}(0.0) Gray{N0f8}(0.0); Gray{N0f8}(0.0) Gray{N0f8}(0.0) … Gray{N0f8}(0.0) Gray{N0f8}(0.0); … ; Gray{N0f8}(0.0) Gray{N0f8}(0.0) … Gray{N0f8}(0.0) Gray{N0f8}(0.0); Gray{N0f8}(0.0) Gray{N0f8}(0.0) … Gray{N0f8}(0.0) Gray{N0f8}(0.0)]\n",
       " [Gray{N0f8}(0.0) Gray{N0f8}(0.0) … Gray{N0f8}(0.0) Gray{N0f8}(0.0); Gray{N0f8}(0.0) Gray{N0f8}(0.0) … Gray{N0f8}(0.0) Gray{N0f8}(0.0); … ; Gray{N0f8}(0.0) Gray{N0f8}(0.0) … Gray{N0f8}(0.0) Gray{N0f8}(0.0); Gray{N0f8}(0.0) Gray{N0f8}(0.0) … Gray{N0f8}(0.0) Gray{N0f8}(0.0)]\n",
       " [Gray{N0f8}(0.0) Gray{N0f8}(0.0) … Gray{N0f8}(0.0) Gray{N0f8}(0.0); Gray{N0f8}(0.0) Gray{N0f8}(0.0) … Gray{N0f8}(0.0) Gray{N0f8}(0.0); … ; Gray{N0f8}(0.0) Gray{N0f8}(0.0) … Gray{N0f8}(0.0) Gray{N0f8}(0.0); Gray{N0f8}(0.0) Gray{N0f8}(0.0) … Gray{N0f8}(0.0) Gray{N0f8}(0.0)]\n",
       " [Gray{N0f8}(0.0) Gray{N0f8}(0.0) … Gray{N0f8}(0.0) Gray{N0f8}(0.0); Gray{N0f8}(0.0) Gray{N0f8}(0.0) … Gray{N0f8}(0.0) Gray{N0f8}(0.0); … ; Gray{N0f8}(0.0) Gray{N0f8}(0.0) … Gray{N0f8}(0.0) Gray{N0f8}(0.0); Gray{N0f8}(0.0) Gray{N0f8}(0.0) … Gray{N0f8}(0.0) Gray{N0f8}(0.0)]\n",
       " [Gray{N0f8}(0.0) Gray{N0f8}(0.0) … Gray{N0f8}(0.0) Gray{N0f8}(0.0); Gray{N0f8}(0.0) Gray{N0f8}(0.0) … Gray{N0f8}(0.0) Gray{N0f8}(0.0); … ; Gray{N0f8}(0.0) Gray{N0f8}(0.0) … Gray{N0f8}(0.0) Gray{N0f8}(0.0); Gray{N0f8}(0.0) Gray{N0f8}(0.0) … Gray{N0f8}(0.0) Gray{N0f8}(0.0)]\n",
       " [Gray{N0f8}(0.0) Gray{N0f8}(0.0) … Gray{N0f8}(0.0) Gray{N0f8}(0.0); Gray{N0f8}(0.0) Gray{N0f8}(0.0) … Gray{N0f8}(0.0) Gray{N0f8}(0.0); … ; Gray{N0f8}(0.0) Gray{N0f8}(0.0) … Gray{N0f8}(0.0) Gray{N0f8}(0.0); Gray{N0f8}(0.0) Gray{N0f8}(0.0) … Gray{N0f8}(0.0) Gray{N0f8}(0.0)]\n",
       " [Gray{N0f8}(0.0) Gray{N0f8}(0.0) … Gray{N0f8}(0.0) Gray{N0f8}(0.0); Gray{N0f8}(0.0) Gray{N0f8}(0.0) … Gray{N0f8}(0.0) Gray{N0f8}(0.0); … ; Gray{N0f8}(0.0) Gray{N0f8}(0.0) … Gray{N0f8}(0.0) Gray{N0f8}(0.0); Gray{N0f8}(0.0) Gray{N0f8}(0.0) … Gray{N0f8}(0.0) Gray{N0f8}(0.0)]\n",
       " [Gray{N0f8}(0.0) Gray{N0f8}(0.0) … Gray{N0f8}(0.0) Gray{N0f8}(0.0); Gray{N0f8}(0.0) Gray{N0f8}(0.0) … Gray{N0f8}(0.0) Gray{N0f8}(0.0); … ; Gray{N0f8}(0.0) Gray{N0f8}(0.0) … Gray{N0f8}(0.0) Gray{N0f8}(0.0); Gray{N0f8}(0.0) Gray{N0f8}(0.0) … Gray{N0f8}(0.0) Gray{N0f8}(0.0)]\n",
       " ⋮                                                                                                                                                                                                                                                                               \n",
       " [Gray{N0f8}(0.0) Gray{N0f8}(0.0) … Gray{N0f8}(0.0) Gray{N0f8}(0.0); Gray{N0f8}(0.0) Gray{N0f8}(0.0) … Gray{N0f8}(0.0) Gray{N0f8}(0.0); … ; Gray{N0f8}(0.0) Gray{N0f8}(0.0) … Gray{N0f8}(0.0) Gray{N0f8}(0.0); Gray{N0f8}(0.0) Gray{N0f8}(0.0) … Gray{N0f8}(0.0) Gray{N0f8}(0.0)]\n",
       " [Gray{N0f8}(0.0) Gray{N0f8}(0.0) … Gray{N0f8}(0.0) Gray{N0f8}(0.0); Gray{N0f8}(0.0) Gray{N0f8}(0.0) … Gray{N0f8}(0.0) Gray{N0f8}(0.0); … ; Gray{N0f8}(0.0) Gray{N0f8}(0.0) … Gray{N0f8}(0.0) Gray{N0f8}(0.0); Gray{N0f8}(0.0) Gray{N0f8}(0.0) … Gray{N0f8}(0.0) Gray{N0f8}(0.0)]\n",
       " [Gray{N0f8}(0.0) Gray{N0f8}(0.0) … Gray{N0f8}(0.0) Gray{N0f8}(0.0); Gray{N0f8}(0.0) Gray{N0f8}(0.0) … Gray{N0f8}(0.0) Gray{N0f8}(0.0); … ; Gray{N0f8}(0.0) Gray{N0f8}(0.0) … Gray{N0f8}(0.0) Gray{N0f8}(0.0); Gray{N0f8}(0.0) Gray{N0f8}(0.0) … Gray{N0f8}(0.0) Gray{N0f8}(0.0)]\n",
       " [Gray{N0f8}(0.0) Gray{N0f8}(0.0) … Gray{N0f8}(0.0) Gray{N0f8}(0.0); Gray{N0f8}(0.0) Gray{N0f8}(0.0) … Gray{N0f8}(0.0) Gray{N0f8}(0.0); … ; Gray{N0f8}(0.0) Gray{N0f8}(0.0) … Gray{N0f8}(0.0) Gray{N0f8}(0.0); Gray{N0f8}(0.0) Gray{N0f8}(0.0) … Gray{N0f8}(0.0) Gray{N0f8}(0.0)]\n",
       " [Gray{N0f8}(0.0) Gray{N0f8}(0.0) … Gray{N0f8}(0.0) Gray{N0f8}(0.0); Gray{N0f8}(0.0) Gray{N0f8}(0.0) … Gray{N0f8}(0.0) Gray{N0f8}(0.0); … ; Gray{N0f8}(0.0) Gray{N0f8}(0.0) … Gray{N0f8}(0.0) Gray{N0f8}(0.0); Gray{N0f8}(0.0) Gray{N0f8}(0.0) … Gray{N0f8}(0.0) Gray{N0f8}(0.0)]\n",
       " [Gray{N0f8}(0.0) Gray{N0f8}(0.0) … Gray{N0f8}(0.0) Gray{N0f8}(0.0); Gray{N0f8}(0.0) Gray{N0f8}(0.0) … Gray{N0f8}(0.0) Gray{N0f8}(0.0); … ; Gray{N0f8}(0.0) Gray{N0f8}(0.0) … Gray{N0f8}(0.0) Gray{N0f8}(0.0); Gray{N0f8}(0.0) Gray{N0f8}(0.0) … Gray{N0f8}(0.0) Gray{N0f8}(0.0)]\n",
       " [Gray{N0f8}(0.0) Gray{N0f8}(0.0) … Gray{N0f8}(0.0) Gray{N0f8}(0.0); Gray{N0f8}(0.0) Gray{N0f8}(0.0) … Gray{N0f8}(0.0) Gray{N0f8}(0.0); … ; Gray{N0f8}(0.0) Gray{N0f8}(0.0) … Gray{N0f8}(0.0) Gray{N0f8}(0.0); Gray{N0f8}(0.0) Gray{N0f8}(0.0) … Gray{N0f8}(0.0) Gray{N0f8}(0.0)]\n",
       " [Gray{N0f8}(0.0) Gray{N0f8}(0.0) … Gray{N0f8}(0.0) Gray{N0f8}(0.0); Gray{N0f8}(0.0) Gray{N0f8}(0.0) … Gray{N0f8}(0.0) Gray{N0f8}(0.0); … ; Gray{N0f8}(0.0) Gray{N0f8}(0.0) … Gray{N0f8}(0.0) Gray{N0f8}(0.0); Gray{N0f8}(0.0) Gray{N0f8}(0.0) … Gray{N0f8}(0.0) Gray{N0f8}(0.0)]\n",
       " [Gray{N0f8}(0.0) Gray{N0f8}(0.0) … Gray{N0f8}(0.0) Gray{N0f8}(0.0); Gray{N0f8}(0.0) Gray{N0f8}(0.0) … Gray{N0f8}(0.0) Gray{N0f8}(0.0); … ; Gray{N0f8}(0.0) Gray{N0f8}(0.0) … Gray{N0f8}(0.0) Gray{N0f8}(0.0); Gray{N0f8}(0.0) Gray{N0f8}(0.0) … Gray{N0f8}(0.0) Gray{N0f8}(0.0)]\n",
       " [Gray{N0f8}(0.0) Gray{N0f8}(0.0) … Gray{N0f8}(0.0) Gray{N0f8}(0.0); Gray{N0f8}(0.0) Gray{N0f8}(0.0) … Gray{N0f8}(0.0) Gray{N0f8}(0.0); … ; Gray{N0f8}(0.0) Gray{N0f8}(0.0) … Gray{N0f8}(0.0) Gray{N0f8}(0.0); Gray{N0f8}(0.0) Gray{N0f8}(0.0) … Gray{N0f8}(0.0) Gray{N0f8}(0.0)]\n",
       " [Gray{N0f8}(0.0) Gray{N0f8}(0.0) … Gray{N0f8}(0.0) Gray{N0f8}(0.0); Gray{N0f8}(0.0) Gray{N0f8}(0.0) … Gray{N0f8}(0.0) Gray{N0f8}(0.0); … ; Gray{N0f8}(0.0) Gray{N0f8}(0.0) … Gray{N0f8}(0.0) Gray{N0f8}(0.0); Gray{N0f8}(0.0) Gray{N0f8}(0.0) … Gray{N0f8}(0.0) Gray{N0f8}(0.0)]\n",
       " [Gray{N0f8}(0.0) Gray{N0f8}(0.0) … Gray{N0f8}(0.0) Gray{N0f8}(0.0); Gray{N0f8}(0.0) Gray{N0f8}(0.0) … Gray{N0f8}(0.0) Gray{N0f8}(0.0); … ; Gray{N0f8}(0.0) Gray{N0f8}(0.0) … Gray{N0f8}(0.0) Gray{N0f8}(0.0); Gray{N0f8}(0.0) Gray{N0f8}(0.0) … Gray{N0f8}(0.0) Gray{N0f8}(0.0)]"
      ]
     },
     "execution_count": 117,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "images_0 = training_images[training_labels .== 0];\n",
    "images_0[1:100]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 52,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "4932"
      ]
     },
     "execution_count": 52,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "length(images_0)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's check that the numerical SVD gives the correct result for the 0 images:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 118,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "vectorize (generic function with 1 method)"
      ]
     },
     "execution_count": 118,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "function vectorize(image) \n",
    "    return round.(Int, 256*image[i].val for i in eachindex(image))\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 119,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 76, 85, 85, 85, 86, 85, 85, 28, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 7, 57, 199, 247, 253, 253, 253, 254, 253, 253, 200, 25, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 38, 179, 253, 254, 253, 231, 224, 146, 226, 250, 253, 253, 129, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 48, 166, 253, 253, 191, 112, 25, 0, 0, 0, 147, 253, 253, 253, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 51, 239, 254, 241, 63, 0, 0, 0, 0, 0, 0, 48, 230, 254, 254, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 10, 164, 239, 253, 234, 71, 0, 0, 0, 0, 0, 0, 0, 29, 216, 253, 253, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 60, 253, 254, 253, 209, 19, 0, 0, 0, 0, 0, 0, 0, 85, 253, 253, 142, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 54, 225, 253, 254, 180, 84, 28, 0, 0, 0, 0, 0, 0, 0, 179, 253, 234, 37, 0, 0, 0, 0, 0, 0, 0, 0, 0, 48, 228, 253, 253, 191, 12, 0, 0, 0, 0, 0, 0, 0, 0, 0, 226, 253, 146, 0, 0, 0, 0, 0, 0, 0, 0, 0, 51, 239, 254, 254, 254, 114, 75, 0, 0, 0, 0, 0, 0, 0, 0, 114, 254, 197, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 160, 253, 253, 253, 253, 254, 121, 0, 0, 0, 0, 0, 0, 0, 48, 239, 224, 131, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 254, 253, 240, 203, 253, 229, 21, 0, 0, 0, 0, 0, 0, 7, 166, 254, 168, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 160, 253, 234, 84, 96, 47, 0, 0, 0, 0, 0, 0, 7, 132, 253, 163, 56, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 50, 238, 253, 253, 190, 79, 0, 0, 0, 0, 0, 0, 136, 253, 174, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 57, 254, 254, 256, 254, 254, 254, 254, 256, 254, 254, 226, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 6, 122, 168, 169, 244, 253, 253, 253, 254, 253, 187, 71, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 50, 166, 196, 196, 197, 149, 12, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]"
     ]
    }
   ],
   "source": [
    "show(vectorize(images[2]))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 55,
   "metadata": {},
   "outputs": [],
   "source": [
    "matrix_0s = reduce(hcat, vectorize.(images_0));"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 120,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "SVD{Float64,Float64,Array{Float64,2}}([-1.11022e-16 -8.32667e-17 … 0.263017 0.108898; 0.0 1.11022e-16 … 0.759198 -0.217008; … ; 0.0 0.0 … 0.0 0.0; 0.0 0.0 … 0.0 0.0], [1.53412e5, 53182.6, 43389.3, 33227.4, 26282.7, 23954.3, 22724.6, 21385.5, 19505.4, 19000.8  …  1.32435e-11, 1.32435e-11, 1.32435e-11, 1.32435e-11, 1.32435e-11, 1.32435e-11, 1.32435e-11, 1.32435e-11, 8.05781e-12, 1.62903e-12], [-0.0146172 -0.0155562 … -0.0130666 -0.00895334; -0.00861048 -0.00649652 … 0.00223801 0.00682512; … ; 0.0 0.649508 … 0.00523472 0.00408817; -0.9741 0.00190592 … 0.00231786 0.000850163])"
      ]
     },
     "execution_count": 120,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "using LinearAlgebra\n",
    "\n",
    "U, Σ, V = svd(matrix_0s)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 121,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "true"
      ]
     },
     "execution_count": 121,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "U * Diagonal(Σ) * V' ≈ matrix_0s"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 122,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "true"
      ]
     },
     "execution_count": 122,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sum(Σ.^2) ≈ sum(matrix_0s.^2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 124,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/svg+xml": [
       "<?xml version=\"1.0\" encoding=\"utf-8\"?>\n",
       "<svg xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\" width=\"600\" height=\"400\" viewBox=\"0 0 2400 1600\">\n",
       "<defs>\n",
       "  <clipPath id=\"clip6300\">\n",
       "    <rect x=\"0\" y=\"0\" width=\"2000\" height=\"2000\"/>\n",
       "  </clipPath>\n",
       "</defs>\n",
       "<defs>\n",
       "  <clipPath id=\"clip6301\">\n",
       "    <rect x=\"0\" y=\"0\" width=\"2400\" height=\"1600\"/>\n",
       "  </clipPath>\n",
       "</defs>\n",
       "<polygon clip-path=\"url(#clip6301)\" points=\"\n",
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       "<circle clip-path=\"url(#clip6303)\" style=\"fill:#000000; stroke:none; fill-opacity:1\" cx=\"2260.55\" cy=\"1462.26\" r=\"18\"/>\n",
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       "1958.43,251.724 2249.26,251.724 2249.26,130.764 1958.43,130.764 \n",
       "  \" fill=\"#ffffff\" fill-rule=\"evenodd\" fill-opacity=\"1\"/>\n",
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       "  \"/>\n",
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       "<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:48px; text-anchor:start;\" transform=\"rotate(0, 2150.43, 208.744)\" x=\"2150.43\" y=\"208.744\">y1</text>\n",
       "</g>\n",
       "</svg>\n"
      ]
     },
     "execution_count": 124,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "using Plots\n",
    "\n",
    "scatter(log10.(Σ))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's look at the \"average 0\":"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 125,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "normalize_image (generic function with 1 method)"
      ]
     },
     "execution_count": 125,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "function normalize_image(image)\n",
    "    image2 = abs.(image)\n",
    "    m = maximum(image2)\n",
    "    image2 = image2 ./ m\n",
    "    return Gray.(reshape(image2, 28, 28))\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 126,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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",
      "text/plain": [
       "28×28 Array{Gray{Float64},2} with eltype Gray{Float64}:\n",
       " Gray{Float64}(0.0)  …  Gray{Float64}(0.0)        \n",
       " Gray{Float64}(0.0)     Gray{Float64}(0.0)        \n",
       " Gray{Float64}(0.0)     Gray{Float64}(0.0)        \n",
       " Gray{Float64}(0.0)     Gray{Float64}(0.0)        \n",
       " Gray{Float64}(0.0)     Gray{Float64}(0.0)        \n",
       " Gray{Float64}(0.0)  …  Gray{Float64}(0.0)        \n",
       " Gray{Float64}(0.0)     Gray{Float64}(7.35929e-5) \n",
       " Gray{Float64}(0.0)     Gray{Float64}(0.000156259)\n",
       " Gray{Float64}(0.0)     Gray{Float64}(0.0)        \n",
       " Gray{Float64}(0.0)     Gray{Float64}(0.0)        \n",
       " Gray{Float64}(0.0)  …  Gray{Float64}(0.0)        \n",
       " Gray{Float64}(0.0)     Gray{Float64}(0.0)        \n",
       " Gray{Float64}(0.0)     Gray{Float64}(0.0)        \n",
       " ⋮                   ⋱                            \n",
       " Gray{Float64}(0.0)     Gray{Float64}(1.20975e-5) \n",
       " Gray{Float64}(0.0)     Gray{Float64}(0.000104845)\n",
       " Gray{Float64}(0.0)     Gray{Float64}(0.0)        \n",
       " Gray{Float64}(0.0)     Gray{Float64}(0.0)        \n",
       " Gray{Float64}(0.0)  …  Gray{Float64}(0.0)        \n",
       " Gray{Float64}(0.0)     Gray{Float64}(3.93167e-5) \n",
       " Gray{Float64}(0.0)     Gray{Float64}(7.25847e-5) \n",
       " Gray{Float64}(0.0)     Gray{Float64}(0.0)        \n",
       " Gray{Float64}(0.0)     Gray{Float64}(0.0)        \n",
       " Gray{Float64}(0.0)  …  Gray{Float64}(0.0)        \n",
       " Gray{Float64}(0.0)     Gray{Float64}(0.0)        \n",
       " Gray{Float64}(0.0)     Gray{Float64}(0.0)        "
      ]
     },
     "execution_count": 126,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "using Statistics\n",
    "\n",
    "normalize_image(mean(matrix_0s, dims=2))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Applying SVD to all images"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 127,
   "metadata": {},
   "outputs": [],
   "source": [
    "big_image_matrix = reduce(hcat, vectorize.(training_images));"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 128,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(784, 50000)"
      ]
     },
     "execution_count": 128,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "size(big_image_matrix)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 129,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "784×50000 Array{Int64,2}:\n",
       " 0  0  0  0  0  0  0  0  0  0  0  0  0  …  0  0  0  0  0  0  0  0  0  0  0  0\n",
       " 0  0  0  0  0  0  0  0  0  0  0  0  0     0  0  0  0  0  0  0  0  0  0  0  0\n",
       " 0  0  0  0  0  0  0  0  0  0  0  0  0     0  0  0  0  0  0  0  0  0  0  0  0\n",
       " 0  0  0  0  0  0  0  0  0  0  0  0  0     0  0  0  0  0  0  0  0  0  0  0  0\n",
       " 0  0  0  0  0  0  0  0  0  0  0  0  0     0  0  0  0  0  0  0  0  0  0  0  0\n",
       " 0  0  0  0  0  0  0  0  0  0  0  0  0  …  0  0  0  0  0  0  0  0  0  0  0  0\n",
       " 0  0  0  0  0  0  0  0  0  0  0  0  0     0  0  0  0  0  0  0  0  0  0  0  0\n",
       " 0  0  0  0  0  0  0  0  0  0  0  0  0     0  0  0  0  0  0  0  0  0  0  0  0\n",
       " 0  0  0  0  0  0  0  0  0  0  0  0  0     0  0  0  0  0  0  0  0  0  0  0  0\n",
       " 0  0  0  0  0  0  0  0  0  0  0  0  0     0  0  0  0  0  0  0  0  0  0  0  0\n",
       " 0  0  0  0  0  0  0  0  0  0  0  0  0  …  0  0  0  0  0  0  0  0  0  0  0  0\n",
       " 0  0  0  0  0  0  0  0  0  0  0  0  0     0  0  0  0  0  0  0  0  0  0  0  0\n",
       " 0  0  0  0  0  0  0  0  0  0  0  0  0     0  0  0  0  0  0  0  0  0  0  0  0\n",
       " ⋮              ⋮              ⋮        ⋱        ⋮              ⋮            \n",
       " 0  0  0  0  0  0  0  0  0  0  0  0  0     0  0  0  0  0  0  0  0  0  0  0  0\n",
       " 0  0  0  0  0  0  0  0  0  0  0  0  0     0  0  0  0  0  0  0  0  0  0  0  0\n",
       " 0  0  0  0  0  0  0  0  0  0  0  0  0     0  0  0  0  0  0  0  0  0  0  0  0\n",
       " 0  0  0  0  0  0  0  0  0  0  0  0  0  …  0  0  0  0  0  0  0  0  0  0  0  0\n",
       " 0  0  0  0  0  0  0  0  0  0  0  0  0     0  0  0  0  0  0  0  0  0  0  0  0\n",
       " 0  0  0  0  0  0  0  0  0  0  0  0  0     0  0  0  0  0  0  0  0  0  0  0  0\n",
       " 0  0  0  0  0  0  0  0  0  0  0  0  0     0  0  0  0  0  0  0  0  0  0  0  0\n",
       " 0  0  0  0  0  0  0  0  0  0  0  0  0     0  0  0  0  0  0  0  0  0  0  0  0\n",
       " 0  0  0  0  0  0  0  0  0  0  0  0  0  …  0  0  0  0  0  0  0  0  0  0  0  0\n",
       " 0  0  0  0  0  0  0  0  0  0  0  0  0     0  0  0  0  0  0  0  0  0  0  0  0\n",
       " 0  0  0  0  0  0  0  0  0  0  0  0  0     0  0  0  0  0  0  0  0  0  0  0  0\n",
       " 0  0  0  0  0  0  0  0  0  0  0  0  0     0  0  0  0  0  0  0  0  0  0  0  0"
      ]
     },
     "execution_count": 129,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "big_image_matrix"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Extract data for each digit into separate matrices"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 130,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "10-element Array{Array{Int64,2},1}:\n",
       " [0 0 … 0 0; 0 0 … 0 0; … ; 0 0 … 0 0; 0 0 … 0 0]\n",
       " [0 0 … 0 0; 0 0 … 0 0; … ; 0 0 … 0 0; 0 0 … 0 0]\n",
       " [0 0 … 0 0; 0 0 … 0 0; … ; 0 0 … 0 0; 0 0 … 0 0]\n",
       " [0 0 … 0 0; 0 0 … 0 0; … ; 0 0 … 0 0; 0 0 … 0 0]\n",
       " [0 0 … 0 0; 0 0 … 0 0; … ; 0 0 … 0 0; 0 0 … 0 0]\n",
       " [0 0 … 0 0; 0 0 … 0 0; … ; 0 0 … 0 0; 0 0 … 0 0]\n",
       " [0 0 … 0 0; 0 0 … 0 0; … ; 0 0 … 0 0; 0 0 … 0 0]\n",
       " [0 0 … 0 0; 0 0 … 0 0; … ; 0 0 … 0 0; 0 0 … 0 0]\n",
       " [0 0 … 0 0; 0 0 … 0 0; … ; 0 0 … 0 0; 0 0 … 0 0]\n",
       " [0 0 … 0 0; 0 0 … 0 0; … ; 0 0 … 0 0; 0 0 … 0 0]"
      ]
     },
     "execution_count": 130,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "single_digits = [ big_image_matrix[:, training_labels .== i] for i in 0:9 ]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 131,
   "metadata": {},
   "outputs": [],
   "source": [
    "Z = svd.(single_digits);"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 67,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "10-element Array{Array{Float64,2},1}:\n",
       " [-1.11022e-16 -8.32667e-17 … 0.263017 0.108898; 0.0 1.11022e-16 … 0.759198 -0.217008; … ; 0.0 0.0 … 0.0 0.0; 0.0 0.0 … 0.0 0.0]      \n",
       " [3.33067e-16 2.22045e-16 … 0.12277 0.243441; 0.0 1.66533e-16 … 0.669715 0.0243274; … ; 0.0 0.0 … 0.0 0.0; 0.0 0.0 … 0.0 0.0]         \n",
       " [2.22045e-16 -5.55112e-17 … 0.00295705 -0.607003; 0.0 1.94289e-16 … -0.518739 -0.190468; … ; 0.0 0.0 … 0.0 0.0; 0.0 0.0 … 0.0 0.0]   \n",
       " [-1.11022e-16 -5.55112e-17 … 0.217468 -0.467104; 0.0 -2.77556e-16 … 0.00632763 0.044892; … ; 0.0 0.0 … 0.0 0.0; 0.0 0.0 … 0.0 0.0]   \n",
       " [1.11022e-16 0.0 … -0.0842133 -0.364336; 0.0 1.249e-16 … -0.00156071 0.0946388; … ; 0.0 0.0 … 0.0 0.0; 0.0 0.0 … 0.0 0.0]            \n",
       " [-3.33067e-16 -1.38778e-17 … -0.0427051 0.0213372; 0.0 -8.67362e-18 … 0.00325293 -0.281373; … ; 0.0 0.0 … 0.0 0.0; 0.0 0.0 … 0.0 0.0]\n",
       " [-1.11022e-16 -6.245e-17 … 0.413816 -0.033716; 0.0 3.46945e-17 … 0.0728744 -0.229194; … ; 0.0 0.0 … 0.0 0.0; 0.0 0.0 … 0.0 0.0]      \n",
       " [0.0 1.66533e-16 … -0.0205796 0.59204; 0.0 -1.66533e-16 … 0.0452746 -0.58315; … ; 0.0 0.0 … 0.0 0.0; 0.0 0.0 … 0.0 0.0]              \n",
       " [2.22045e-16 5.55112e-17 … 0.272366 -0.0480318; 0.0 -2.77556e-16 … -0.522072 -0.261654; … ; 0.0 0.0 … 0.0 0.0; 0.0 0.0 … 0.0 0.0]    \n",
       " [1.11022e-16 -1.94289e-16 … 0.371535 -0.288321; 0.0 2.22045e-16 … -0.532185 -0.089791; … ; 0.0 0.0 … 0.0 0.0; 0.0 0.0 … 0.0 0.0]     "
      ]
     },
     "execution_count": 67,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Us = first.(Z)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 68,
   "metadata": {},
   "outputs": [],
   "source": [
    "using Statistics"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 69,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "784-element Array{Float64,1}:\n",
       " -1.1102230246251565e-16\n",
       "  0.0                   \n",
       "  0.0                   \n",
       "  0.0                   \n",
       "  0.0                   \n",
       "  0.0                   \n",
       "  0.0                   \n",
       "  0.0                   \n",
       "  0.0                   \n",
       "  0.0                   \n",
       "  0.0                   \n",
       "  0.0                   \n",
       "  0.0                   \n",
       "  ⋮                     \n",
       " -4.4300954484669486e-7 \n",
       " -3.839416055338021e-6  \n",
       "  0.0                   \n",
       "  0.0                   \n",
       "  0.0                   \n",
       " -3.919018919865072e-6  \n",
       " -7.235111852058591e-6  \n",
       "  0.0                   \n",
       "  0.0                   \n",
       "  0.0                   \n",
       "  0.0                   \n",
       "  0.0                   "
      ]
     },
     "execution_count": 69,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "U[:, 1]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 70,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1.0000000000000004"
      ]
     },
     "execution_count": 70,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "norm(U[:, 1])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 71,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
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     ]
    }
   ],
   "source": [
    "show(U[:, 1])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Thresholding an image:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 132,
   "metadata": {},
   "outputs": [
    {
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       "]},&quot;changes&quot;:[(function (val){return (val!=this.model[&quot;changes&quot;]()) ? (this.valueFromJulia[&quot;changes&quot;]=true, this.model[&quot;changes&quot;](val)) : undefined})],&quot;formatted_value&quot;:[(function (val){return (val!=this.model[&quot;formatted_value&quot;]()) ? (this.valueFromJulia[&quot;formatted_value&quot;]=true, this.model[&quot;formatted_value&quot;](val)) : undefined})],&quot;index&quot;:[(function (val){return (val!=this.model[&quot;index&quot;]()) ? (this.valueFromJulia[&quot;index&quot;]=true, this.model[&quot;index&quot;](val)) : undefined}),(function (val){return WebIO.setval({&quot;name&quot;:&quot;formatted_value&quot;,&quot;scope&quot;:&quot;knockout-component-2059f24d-c0d1-477e-bd3d-0098448f7880&quot;,&quot;id&quot;:&quot;ob_88&quot;,&quot;type&quot;:&quot;observable&quot;},WebIO.getval({&quot;name&quot;:&quot;formatted_vals&quot;,&quot;scope&quot;:&quot;knockout-component-2059f24d-c0d1-477e-bd3d-0098448f7880&quot;,&quot;id&quot;:&quot;ob_87&quot;,&quot;type&quot;:&quot;observable&quot;})[(WebIO.getval({&quot;name&quot;:&quot;index&quot;,&quot;scope&quot;:&quot;knockout-component-2059f24d-c0d1-477e-bd3d-0098448f7880&quot;,&quot;id&quot;:&quot;ob_86&quot;,&quot;type&quot;:&quot;observable&quot;})-1)])})]},&quot;systemjs_options&quot;:null,&quot;observables&quot;:{&quot;formatted_vals&quot;:{&quot;sync&quot;:false,&quot;id&quot;:&quot;ob_87&quot;,&quot;value&quot;:[&quot;0.0&quot;,&quot;0.01&quot;,&quot;0.02&quot;,&quot;0.03&quot;,&quot;0.04&quot;,&quot;0.05&quot;,&quot;0.06&quot;,&quot;0.07&quot;,&quot;0.08&quot;,&quot;0.09&quot;,&quot;0.1&quot;,&quot;0.11&quot;,&quot;0.12&quot;]},&quot;changes&quot;:{&quot;sync&quot;:false,&quot;id&quot;:&quot;ob_89&quot;,&quot;value&quot;:0},&quot;formatted_value&quot;:{&quot;sync&quot;:false,&quot;id&quot;:&quot;ob_88&quot;,&quot;value&quot;:&quot;0.06&quot;},&quot;index&quot;:{&quot;sync&quot;:true,&quot;id&quot;:&quot;ob_86&quot;,&quot;value&quot;:7}}},&quot;children&quot;:[{&quot;props&quot;:{&quot;attributes&quot;:{&quot;class&quot;:&quot;interact-flex-row&quot;}},&quot;nodeType&quot;:&quot;DOM&quot;,&quot;type&quot;:&quot;node&quot;,&quot;instanceArgs&quot;:{&quot;namespace&quot;:&quot;html&quot;,&quot;tag&quot;:&quot;div&quot;},&quot;children&quot;:[{&quot;props&quot;:{&quot;attributes&quot;:{&quot;class&quot;:&quot;interact-flex-row-left&quot;}},&quot;nodeType&quot;:&quot;DOM&quot;,&quot;type&quot;:&quot;node&quot;,&quot;instanceArgs&quot;:{&quot;namespace&quot;:&quot;html&quot;,&quot;tag&quot;:&quot;div&quot;},&quot;children&quot;:[{&quot;props&quot;:{&quot;className&quot;:&quot;interact &quot;,&quot;style&quot;:{&quot;padding&quot;:&quot;5px 10px 0px 10px&quot;}},&quot;nodeType&quot;:&quot;DOM&quot;,&quot;type&quot;:&quot;node&quot;,&quot;instanceArgs&quot;:{&quot;namespace&quot;:&quot;html&quot;,&quot;tag&quot;:&quot;label&quot;},&quot;children&quot;:[&quot;θ&quot;]}]},{&quot;props&quot;:{&quot;attributes&quot;:{&quot;class&quot;:&quot;interact-flex-row-center&quot;}},&quot;nodeType&quot;:&quot;DOM&quot;,&quot;type&quot;:&quot;node&quot;,&quot;instanceArgs&quot;:{&quot;namespace&quot;:&quot;html&quot;,&quot;tag&quot;:&quot;div&quot;},&quot;children&quot;:[{&quot;props&quot;:{&quot;max&quot;:13,&quot;min&quot;:1,&quot;attributes&quot;:{&quot;type&quot;:&quot;range&quot;,&quot;data-bind&quot;:&quot;numericValue: index, valueUpdate: &#39;input&#39;, event: {change : function () {this.changes(this.changes()+1)}}&quot;,&quot;orient&quot;:&quot;horizontal&quot;},&quot;step&quot;:1,&quot;className&quot;:&quot;slider slider is-fullwidth&quot;,&quot;style&quot;:{}},&quot;nodeType&quot;:&quot;DOM&quot;,&quot;type&quot;:&quot;node&quot;,&quot;instanceArgs&quot;:{&quot;namespace&quot;:&quot;html&quot;,&quot;tag&quot;:&quot;input&quot;},&quot;children&quot;:[]}]},{&quot;props&quot;:{&quot;attributes&quot;:{&quot;class&quot;:&quot;interact-flex-row-right&quot;}},&quot;nodeType&quot;:&quot;DOM&quot;,&quot;type&quot;:&quot;node&quot;,&quot;instanceArgs&quot;:{&quot;namespace&quot;:&quot;html&quot;,&quot;tag&quot;:&quot;div&quot;},&quot;children&quot;:[{&quot;props&quot;:{&quot;attributes&quot;:{&quot;data-bind&quot;:&quot;text: formatted_value&quot;}},&quot;nodeType&quot;:&quot;DOM&quot;,&quot;type&quot;:&quot;node&quot;,&quot;instanceArgs&quot;:{&quot;namespace&quot;:&quot;html&quot;,&quot;tag&quot;:&quot;p&quot;},&quot;children&quot;:[]}]}]}]}]},{&quot;props&quot;:{},&quot;nodeType&quot;:&quot;Scope&quot;,&quot;type&quot;:&quot;node&quot;,&quot;instanceArgs&quot;:{&quot;imports&quot;:{&quot;data&quot;:[],&quot;type&quot;:&quot;async_block&quot;},&quot;id&quot;:&quot;scope-d55fc8ae-5b58-4f5d-80d4-7f849a764536&quot;,&quot;handlers&quot;:{&quot;obs-output&quot;:[function (updated_htmlstr) {\n",
       "    var el = this.dom.querySelector(&quot;#out&quot;);\n",
       "    WebIO.propUtils.setInnerHtml(el, updated_htmlstr);\n",
       "}]},&quot;systemjs_options&quot;:null,&quot;observables&quot;:{&quot;obs-output&quot;:{&quot;sync&quot;:false,&quot;id&quot;:&quot;ob_92&quot;,&quot;value&quot;:&quot;&lt;div class=&#39;display:none&#39;&gt;&lt;/div&gt;&lt;unsafe-script style=&#39;display:none&#39;&gt;\\nWebIO.mount(this.previousSibling,{&amp;quot;props&amp;quot;:{&amp;quot;attributes&amp;quot;:{&amp;quot;class&amp;quot;:&amp;quot;interact-flex-row&amp;quot;}},&amp;quot;nodeType&amp;quot;:&amp;quot;DOM&amp;quot;,&amp;quot;type&amp;quot;:&amp;quot;node&amp;quot;,&amp;quot;instanceArgs&amp;quot;:{&amp;quot;namespace&amp;quot;:&amp;quot;html&amp;quot;,&amp;quot;tag&amp;quot;:&amp;quot;div&amp;quot;},&amp;quot;children&amp;quot;:[{&amp;quot;props&amp;quot;:{&amp;quot;setInnerHtml&amp;quot;:&amp;quot;&amp;lt;img src=&amp;#39;data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAHAAAABwAQAAAADum8nmAAAABGdBTUEAALGPC/xhBQAAACBjSFJNAAB6JgAAgIQAAPoAAACA6AAAdTAAAOpgAAA6mAAAF3CculE8AAAAAmJLR0QAAd2KE6QAAABgSURBVDjL7dTBDUAhCANQNuj+W7JBlZgYk3+x5ejn9i4NKhjxl1qgTZB0OcX0WElHlsTKMYlqAR5Xyj6DxrSJFqPNMLgy0GH6NG+d+9lkzgHFMd0q7U0Ifhfymp1f46EaAP4OxPiWA2wAAAAASUVORK5CYII=&amp;#39;&amp;gt;&amp;lt;/img&amp;gt;&amp;quot;},&amp;quot;nodeType&amp;quot;:&amp;quot;DOM&amp;quot;,&amp;quot;type&amp;quot;:&amp;quot;node&amp;quot;,&amp;quot;instanceArgs&amp;quot;:{&amp;quot;namespace&amp;quot;:&amp;quot;html&amp;quot;,&amp;quot;tag&amp;quot;:&amp;quot;div&amp;quot;},&amp;quot;children&amp;quot;:[]}]})&lt;/unsafe-script&gt;&quot;}}},&quot;children&quot;:[{&quot;props&quot;:{&quot;id&quot;:&quot;out&quot;,&quot;setInnerHtml&quot;:&quot;&lt;div class=&#39;display:none&#39;&gt;&lt;/div&gt;&lt;unsafe-script style=&#39;display:none&#39;&gt;\\nWebIO.mount(this.previousSibling,{&amp;quot;props&amp;quot;:{&amp;quot;attributes&amp;quot;:{&amp;quot;class&amp;quot;:&amp;quot;interact-flex-row&amp;quot;}},&amp;quot;nodeType&amp;quot;:&amp;quot;DOM&amp;quot;,&amp;quot;type&amp;quot;:&amp;quot;node&amp;quot;,&amp;quot;instanceArgs&amp;quot;:{&amp;quot;namespace&amp;quot;:&amp;quot;html&amp;quot;,&amp;quot;tag&amp;quot;:&amp;quot;div&amp;quot;},&amp;quot;children&amp;quot;:[{&amp;quot;props&amp;quot;:{&amp;quot;setInnerHtml&amp;quot;:&amp;quot;&amp;lt;img src=&amp;#39;data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAHAAAABwAQAAAADum8nmAAAABGdBTUEAALGPC/xhBQAAACBjSFJNAAB6JgAAgIQAAPoAAACA6AAAdTAAAOpgAAA6mAAAF3CculE8AAAAAmJLR0QAAd2KE6QAAABgSURBVDjL7dTBDUAhCANQNuj+W7JBlZgYk3+x5ejn9i4NKhjxl1qgTZB0OcX0WElHlsTKMYlqAR5Xyj6DxrSJFqPNMLgy0GH6NG+d+9lkzgHFMd0q7U0Ifhfymp1f46EaAP4OxPiWA2wAAAAASUVORK5CYII=&amp;#39;&amp;gt;&amp;lt;/img&amp;gt;&amp;quot;},&amp;quot;nodeType&amp;quot;:&amp;quot;DOM&amp;quot;,&amp;quot;type&amp;quot;:&amp;quot;node&amp;quot;,&amp;quot;instanceArgs&amp;quot;:{&amp;quot;namespace&amp;quot;:&amp;quot;html&amp;quot;,&amp;quot;tag&amp;quot;:&amp;quot;div&amp;quot;},&amp;quot;children&amp;quot;:[]}]})&lt;/unsafe-script&gt;&quot;},&quot;nodeType&quot;:&quot;DOM&quot;,&quot;type&quot;:&quot;node&quot;,&quot;instanceArgs&quot;:{&quot;namespace&quot;:&quot;html&quot;,&quot;tag&quot;:&quot;div&quot;},&quot;children&quot;:[]}]}]})</unsafe-script>\n",
       "</div>"
      ],
      "text/plain": [
       "Widget{:manipulate,Any}(OrderedDict{Symbol,Any}(:θ=>Widget{:slider,Float64}(OrderedDict{Symbol,Any}(:changes=>Observable{Int64} with 1 listeners. Value:\n",
       "0,:index=>Observable{Any} with 2 listeners. Value:\n",
       "7,:formatted_vals=>Observable{Any} with 1 listeners. Value:\n",
       "[\"0.0\", \"0.01\", \"0.02\", \"0.03\", \"0.04\", \"0.05\", \"0.06\", \"0.07\", \"0.08\", \"0.09\", \"0.1\", \"0.11\", \"0.12\"],:formatted_value=>Observable{String} with 1 listeners. Value:\n",
       "\"0.06\",:value=>Observable{Float64} with 2 listeners. Value:\n",
       "0.06), Observable{Float64} with 2 listeners. Value:\n",
       "0.06, Scope(\"knockout-component-2059f24d-c0d1-477e-bd3d-0098448f7880\", Node{DOM}(DOM(:html, :div), Any[Node{DOM}(DOM(:html, :div), Any[Node{DOM}(DOM(:html, :label), Any[\"θ\"], Dict{Symbol,Any}(:className=>\"interact \",:style=>Dict{Any,Any}(:padding=>\"5px 10px 0px 10px\")), 1)], Dict{Symbol,Any}(:attributes=>Dict(\"class\"=>\"interact-flex-row-left\")), 2), Node{DOM}(DOM(:html, :div), Any[Node{DOM}(DOM(:html, :input), Any[], Dict{Symbol,Any}(:max=>13,:min=>1,:attributes=>Dict{Any,Any}(:type=>\"range\",Symbol(\"data-bind\")=>\"numericValue: index, valueUpdate: 'input', event: {change : function () {this.changes(this.changes()+1)}}\",\"orient\"=>\"horizontal\"),:step=>1,:className=>\"slider slider is-fullwidth\",:style=>Dict{Any,Any}()), 0)], Dict{Symbol,Any}(:attributes=>Dict(\"class\"=>\"interact-flex-row-center\")), 1), Node{DOM}(DOM(:html, :div), Any[Node{DOM}(DOM(:html, :p), Any[], Dict{Symbol,Any}(:attributes=>Dict(\"data-bind\"=>\"text: formatted_value\")), 0)], Dict{Symbol,Any}(:attributes=>Dict(\"class\"=>\"interact-flex-row-right\")), 1)], Dict{Symbol,Any}(:attributes=>Dict(\"class\"=>\"interact-flex-row\")), 7), Dict{String,Tuple{Observables.AbstractObservable,Union{Nothing, Bool}}}(\"formatted_vals\"=>(Observable{Any} with 1 listeners. Value:\n",
       "[\"0.0\", \"0.01\", \"0.02\", \"0.03\", \"0.04\", \"0.05\", \"0.06\", \"0.07\", \"0.08\", \"0.09\", \"0.1\", \"0.11\", \"0.12\"], nothing),\"changes\"=>(Observable{Int64} with 1 listeners. Value:\n",
       "0, nothing),\"formatted_value\"=>(Observable{String} with 1 listeners. Value:\n",
       "\"0.06\", nothing),\"index\"=>(Observable{Any} with 2 listeners. Value:\n",
       "7, nothing)), Set(String[]), nothing, Any[\"knockout\"=>\"/Users/alanedelman/.julia/packages/Knockout/JIqpG/src/../assets/knockout.js\", \"knockout_punches\"=>\"/Users/alanedelman/.julia/packages/Knockout/JIqpG/src/../assets/knockout_punches.js\", \"/Users/alanedelman/.julia/packages/InteractBase/PTCUD/src/../assets/all.js\", \"/Users/alanedelman/.julia/packages/InteractBase/PTCUD/src/../assets/style.css\", \"/Users/alanedelman/.julia/packages/InteractBulma/PH56C/src/../assets/main.css\"], Dict{Any,Any}(\"formatted_vals\"=>Any[JSString(\"(function (val){return (val!=this.model[\\\"formatted_vals\\\"]()) ? (this.valueFromJulia[\\\"formatted_vals\\\"]=true, this.model[\\\"formatted_vals\\\"](val)) : undefined})\")],\"_promises\"=>Dict{Any,Any}(\"importsLoaded\"=>Any[JSString(\"function (ko, koPunches) {\\n    ko.punches.enableAll();\\n    ko.bindingHandlers.numericValue = {\\n        init : function(element, valueAccessor, allBindings, data, context) {\\n            var stringified = ko.observable(ko.unwrap(valueAccessor()));\\n            stringified.subscribe(function(value) {\\n                var val = parseFloat(value);\\n                if (!isNaN(val)) {\\n                    valueAccessor()(val);\\n                }\\n            })\\n            valueAccessor().subscribe(function(value) {\\n                var str = JSON.stringify(value);\\n                if ((str == \\\"0\\\") && ([\\\"-0\\\", \\\"-0.\\\"].indexOf(stringified()) >= 0))\\n                     return;\\n                 if ([\\\"null\\\", \\\"\\\"].indexOf(str) >= 0)\\n                     return;\\n                stringified(str);\\n            })\\n            ko.applyBindingsToNode(element, { value: stringified, valueUpdate: allBindings.get('valueUpdate')}, context);\\n        }\\n    };\\n    var json_data = JSON.parse(\\\"{\\\\\\\"formatted_vals\\\\\\\":[\\\\\\\"0.0\\\\\\\",\\\\\\\"0.01\\\\\\\",\\\\\\\"0.02\\\\\\\",\\\\\\\"0.03\\\\\\\",\\\\\\\"0.04\\\\\\\",\\\\\\\"0.05\\\\\\\",\\\\\\\"0.06\\\\\\\",\\\\\\\"0.07\\\\\\\",\\\\\\\"0.08\\\\\\\",\\\\\\\"0.09\\\\\\\",\\\\\\\"0.1\\\\\\\",\\\\\\\"0.11\\\\\\\",\\\\\\\"0.12\\\\\\\"],\\\\\\\"changes\\\\\\\":0,\\\\\\\"formatted_value\\\\\\\":\\\\\\\"0.06\\\\\\\",\\\\\\\"index\\\\\\\":7}\\\");\\n    var self = this;\\n    function AppViewModel() {\\n        for (var key in json_data) {\\n            var el = json_data[key];\\n            this[key] = Array.isArray(el) ? ko.observableArray(el) : ko.observable(el);\\n        }\\n        \\n        \\n        [this[\\\"formatted_vals\\\"].subscribe((function (val){!(this.valueFromJulia[\\\"formatted_vals\\\"]) ? (WebIO.setval({\\\"name\\\":\\\"formatted_vals\\\",\\\"scope\\\":\\\"knockout-component-2059f24d-c0d1-477e-bd3d-0098448f7880\\\",\\\"id\\\":\\\"ob_87\\\",\\\"type\\\":\\\"observable\\\"},val)) : undefined; return this.valueFromJulia[\\\"formatted_vals\\\"]=false}),self),this[\\\"changes\\\"].subscribe((function (val){!(this.valueFromJulia[\\\"changes\\\"]) ? (WebIO.setval({\\\"name\\\":\\\"changes\\\",\\\"scope\\\":\\\"knockout-component-2059f24d-c0d1-477e-bd3d-0098448f7880\\\",\\\"id\\\":\\\"ob_89\\\",\\\"type\\\":\\\"observable\\\"},val)) : undefined; return this.valueFromJulia[\\\"changes\\\"]=false}),self),this[\\\"formatted_value\\\"].subscribe((function (val){!(this.valueFromJulia[\\\"formatted_value\\\"]) ? (WebIO.setval({\\\"name\\\":\\\"formatted_value\\\",\\\"scope\\\":\\\"knockout-component-2059f24d-c0d1-477e-bd3d-0098448f7880\\\",\\\"id\\\":\\\"ob_88\\\",\\\"type\\\":\\\"observable\\\"},val)) : undefined; return this.valueFromJulia[\\\"formatted_value\\\"]=false}),self),this[\\\"index\\\"].subscribe((function (val){!(this.valueFromJulia[\\\"index\\\"]) ? (WebIO.setval({\\\"name\\\":\\\"index\\\",\\\"scope\\\":\\\"knockout-component-2059f24d-c0d1-477e-bd3d-0098448f7880\\\",\\\"id\\\":\\\"ob_86\\\",\\\"type\\\":\\\"observable\\\"},val)) : undefined; return this.valueFromJulia[\\\"index\\\"]=false}),self)]\\n        \\n    }\\n    self.model = new AppViewModel();\\n    self.valueFromJulia = {};\\n    for (var key in json_data) {\\n        self.valueFromJulia[key] = false;\\n    }\\n    ko.applyBindings(self.model, self.dom);\\n}\\n\")]),\"changes\"=>Any[JSString(\"(function (val){return (val!=this.model[\\\"changes\\\"]()) ? (this.valueFromJulia[\\\"changes\\\"]=true, this.model[\\\"changes\\\"](val)) : undefined})\")],\"formatted_value\"=>Any[JSString(\"(function (val){return (val!=this.model[\\\"formatted_value\\\"]()) ? (this.valueFromJulia[\\\"formatted_value\\\"]=true, this.model[\\\"formatted_value\\\"](val)) : undefined})\")],\"index\"=>Any[JSString(\"(function (val){return (val!=this.model[\\\"index\\\"]()) ? (this.valueFromJulia[\\\"index\\\"]=true, this.model[\\\"index\\\"](val)) : undefined})\"), JSString(\"(function (val){return WebIO.setval({\\\"name\\\":\\\"formatted_value\\\",\\\"scope\\\":\\\"knockout-component-2059f24d-c0d1-477e-bd3d-0098448f7880\\\",\\\"id\\\":\\\"ob_88\\\",\\\"type\\\":\\\"observable\\\"},WebIO.getval({\\\"name\\\":\\\"formatted_vals\\\",\\\"scope\\\":\\\"knockout-component-2059f24d-c0d1-477e-bd3d-0098448f7880\\\",\\\"id\\\":\\\"ob_87\\\",\\\"type\\\":\\\"observable\\\"})[(WebIO.getval({\\\"name\\\":\\\"index\\\",\\\"scope\\\":\\\"knockout-component-2059f24d-c0d1-477e-bd3d-0098448f7880\\\",\\\"id\\\":\\\"ob_86\\\",\\\"type\\\":\\\"observable\\\"})-1)])})\")]), ConnectionPool(Channel{Any}(sz_max:9223372036854775807,sz_curr:4), Set(AbstractConnection[]), Channel{AbstractConnection}(sz_max:32,sz_curr:0))), ##52#53{#dom#15{##dom#13#14{Dict{Any,Any},DOM}},typeof(scope)}(#dom#15{##dom#13#14{Dict{Any,Any},DOM}}(##dom#13#14{Dict{Any,Any},DOM}(Dict{Any,Any}(:className=>\"field\"), DOM(:html, :div))), scope))), Observable{Any} with 0 listeners. Value:\n",
       "Gray{Bool}[Gray{Bool}(false) Gray{Bool}(false) … Gray{Bool}(false) Gray{Bool}(false); Gray{Bool}(false) Gray{Bool}(false) … Gray{Bool}(false) Gray{Bool}(false); … ; Gray{Bool}(false) Gray{Bool}(false) … Gray{Bool}(false) Gray{Bool}(false); Gray{Bool}(false) Gray{Bool}(false) … Gray{Bool}(false) Gray{Bool}(false)], nothing, getfield(InteractBase, Symbol(\"##142#143\"))())"
      ]
     },
     "execution_count": 132,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "@manipulate for θ in 0.0:0.01:0.12\n",
    "    image = abs.(U[:, 1]) .> θ\n",
    "    Gray.(reshape(image, 28, 28))\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 73,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "10-element Array{Array{Float64,2},1}:\n",
       " [-1.11022e-16 -8.32667e-17 … 0.263017 0.108898; 0.0 1.11022e-16 … 0.759198 -0.217008; … ; 0.0 0.0 … 0.0 0.0; 0.0 0.0 … 0.0 0.0]      \n",
       " [3.33067e-16 2.22045e-16 … 0.12277 0.243441; 0.0 1.66533e-16 … 0.669715 0.0243274; … ; 0.0 0.0 … 0.0 0.0; 0.0 0.0 … 0.0 0.0]         \n",
       " [2.22045e-16 -5.55112e-17 … 0.00295705 -0.607003; 0.0 1.94289e-16 … -0.518739 -0.190468; … ; 0.0 0.0 … 0.0 0.0; 0.0 0.0 … 0.0 0.0]   \n",
       " [-1.11022e-16 -5.55112e-17 … 0.217468 -0.467104; 0.0 -2.77556e-16 … 0.00632763 0.044892; … ; 0.0 0.0 … 0.0 0.0; 0.0 0.0 … 0.0 0.0]   \n",
       " [1.11022e-16 0.0 … -0.0842133 -0.364336; 0.0 1.249e-16 … -0.00156071 0.0946388; … ; 0.0 0.0 … 0.0 0.0; 0.0 0.0 … 0.0 0.0]            \n",
       " [-3.33067e-16 -1.38778e-17 … -0.0427051 0.0213372; 0.0 -8.67362e-18 … 0.00325293 -0.281373; … ; 0.0 0.0 … 0.0 0.0; 0.0 0.0 … 0.0 0.0]\n",
       " [-1.11022e-16 -6.245e-17 … 0.413816 -0.033716; 0.0 3.46945e-17 … 0.0728744 -0.229194; … ; 0.0 0.0 … 0.0 0.0; 0.0 0.0 … 0.0 0.0]      \n",
       " [0.0 1.66533e-16 … -0.0205796 0.59204; 0.0 -1.66533e-16 … 0.0452746 -0.58315; … ; 0.0 0.0 … 0.0 0.0; 0.0 0.0 … 0.0 0.0]              \n",
       " [2.22045e-16 5.55112e-17 … 0.272366 -0.0480318; 0.0 -2.77556e-16 … -0.522072 -0.261654; … ; 0.0 0.0 … 0.0 0.0; 0.0 0.0 … 0.0 0.0]    \n",
       " [1.11022e-16 -1.94289e-16 … 0.371535 -0.288321; 0.0 2.22045e-16 … -0.532185 -0.089791; … ; 0.0 0.0 … 0.0 0.0; 0.0 0.0 … 0.0 0.0]     "
      ]
     },
     "execution_count": 73,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Us"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's look at all the singular vectors for all images:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 74,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<div class='tex2jax_ignore interactbulma'>\n",
       "<div class='display:none'></div><unsafe-script style='display:none'>\n",
       "WebIO.mount(this.previousSibling,{&quot;props&quot;:{},&quot;nodeType&quot;:&quot;DOM&quot;,&quot;type&quot;:&quot;node&quot;,&quot;instanceArgs&quot;:{&quot;namespace&quot;:&quot;html&quot;,&quot;tag&quot;:&quot;div&quot;},&quot;children&quot;:[{&quot;props&quot;:{&quot;className&quot;:&quot;field&quot;},&quot;nodeType&quot;:&quot;DOM&quot;,&quot;type&quot;:&quot;node&quot;,&quot;instanceArgs&quot;:{&quot;namespace&quot;:&quot;html&quot;,&quot;tag&quot;:&quot;div&quot;},&quot;children&quot;:[{&quot;props&quot;:{},&quot;nodeType&quot;:&quot;Scope&quot;,&quot;type&quot;:&quot;node&quot;,&quot;instanceArgs&quot;:{&quot;imports&quot;:{&quot;data&quot;:[{&quot;name&quot;:&quot;knockout&quot;,&quot;type&quot;:&quot;js&quot;,&quot;url&quot;:&quot;/assetserver/06c67466ebc6c52906ffd4101b7c74a47f9567bc-knockout.js&quot;},{&quot;name&quot;:&quot;knockout_punches&quot;,&quot;type&quot;:&quot;js&quot;,&quot;url&quot;:&quot;/assetserver/46b0ee2b2af2730ca29ddb46f091ac108fb8f609-knockout_punches.js&quot;},{&quot;name&quot;:null,&quot;type&quot;:&quot;js&quot;,&quot;url&quot;:&quot;/assetserver/e923a2cf7cc8bb183f572d980ae606c5f08fa0dc-all.js&quot;},{&quot;name&quot;:null,&quot;type&quot;:&quot;css&quot;,&quot;url&quot;:&quot;/assetserver/b973c333a7d1e5a6ca74f1442a2f26b5e4d0580e-style.css&quot;},{&quot;name&quot;:null,&quot;type&quot;:&quot;css&quot;,&quot;url&quot;:&quot;/assetserver/885f9c98e35e2ecc817c654a867b4570a606ec22-main.css&quot;}],&quot;type&quot;:&quot;async_block&quot;},&quot;id&quot;:&quot;knockout-component-2ea6b4cf-0c61-4dba-9552-e16c84fdae9b&quot;,&quot;handlers&quot;:{&quot;_promises&quot;:{&quot;importsLoaded&quot;:[function (ko, koPunches) {\n",
       "    ko.punches.enableAll();\n",
       "    ko.bindingHandlers.numericValue = {\n",
       "        init : function(element, valueAccessor, allBindings, data, context) {\n",
       "            var stringified = ko.observable(ko.unwrap(valueAccessor()));\n",
       "            stringified.subscribe(function(value) {\n",
       "                var val = parseFloat(value);\n",
       "                if (!isNaN(val)) {\n",
       "                    valueAccessor()(val);\n",
       "                }\n",
       "            })\n",
       "            valueAccessor().subscribe(function(value) {\n",
       "                var str = JSON.stringify(value);\n",
       "                if ((str == &quot;0&quot;) &amp;&amp; ([&quot;-0&quot;, &quot;-0.&quot;].indexOf(stringified()) &gt;= 0))\n",
       "                     return;\n",
       "                 if ([&quot;null&quot;, &quot;&quot;].indexOf(str) &gt;= 0)\n",
       "                     return;\n",
       "                stringified(str);\n",
       "            })\n",
       "            ko.applyBindingsToNode(element, { value: stringified, valueUpdate: allBindings.get(&#39;valueUpdate&#39;)}, context);\n",
       "        }\n",
       "    };\n",
       "    var json_data = JSON.parse(&quot;{\\&quot;changes\\&quot;:0,\\&quot;value\\&quot;:0}&quot;);\n",
       "    var self = this;\n",
       "    function AppViewModel() {\n",
       "        for (var key in json_data) {\n",
       "            var el = json_data[key];\n",
       "            this[key] = Array.isArray(el) ? ko.observableArray(el) : ko.observable(el);\n",
       "        }\n",
       "        \n",
       "        \n",
       "        [this[&quot;changes&quot;].subscribe((function (val){!(this.valueFromJulia[&quot;changes&quot;]) ? (WebIO.setval({&quot;name&quot;:&quot;changes&quot;,&quot;scope&quot;:&quot;knockout-component-2ea6b4cf-0c61-4dba-9552-e16c84fdae9b&quot;,&quot;id&quot;:&quot;ob_59&quot;,&quot;type&quot;:&quot;observable&quot;},val)) : undefined; return this.valueFromJulia[&quot;changes&quot;]=false}),self),this[&quot;value&quot;].subscribe((function (val){!(this.valueFromJulia[&quot;value&quot;]) ? (WebIO.setval({&quot;name&quot;:&quot;value&quot;,&quot;scope&quot;:&quot;knockout-component-2ea6b4cf-0c61-4dba-9552-e16c84fdae9b&quot;,&quot;id&quot;:&quot;ob_58&quot;,&quot;type&quot;:&quot;observable&quot;},val)) : undefined; return this.valueFromJulia[&quot;value&quot;]=false}),self)]\n",
       "        \n",
       "    }\n",
       "    self.model = new AppViewModel();\n",
       "    self.valueFromJulia = {};\n",
       "    for (var key in json_data) {\n",
       "        self.valueFromJulia[key] = false;\n",
       "    }\n",
       "    ko.applyBindings(self.model, self.dom);\n",
       "}\n",
       "]},&quot;changes&quot;:[(function (val){return (val!=this.model[&quot;changes&quot;]()) ? (this.valueFromJulia[&quot;changes&quot;]=true, this.model[&quot;changes&quot;](val)) : undefined})],&quot;value&quot;:[(function (val){return (val!=this.model[&quot;value&quot;]()) ? (this.valueFromJulia[&quot;value&quot;]=true, this.model[&quot;value&quot;](val)) : undefined})]},&quot;systemjs_options&quot;:null,&quot;observables&quot;:{&quot;changes&quot;:{&quot;sync&quot;:false,&quot;id&quot;:&quot;ob_59&quot;,&quot;value&quot;:0},&quot;value&quot;:{&quot;sync&quot;:true,&quot;id&quot;:&quot;ob_58&quot;,&quot;value&quot;:0}}},&quot;children&quot;:[{&quot;props&quot;:{&quot;attributes&quot;:{&quot;class&quot;:&quot;interact-flex-row&quot;}},&quot;nodeType&quot;:&quot;DOM&quot;,&quot;type&quot;:&quot;node&quot;,&quot;instanceArgs&quot;:{&quot;namespace&quot;:&quot;html&quot;,&quot;tag&quot;:&quot;div&quot;},&quot;children&quot;:[{&quot;props&quot;:{&quot;attributes&quot;:{&quot;class&quot;:&quot;interact-flex-row-left&quot;}},&quot;nodeType&quot;:&quot;DOM&quot;,&quot;type&quot;:&quot;node&quot;,&quot;instanceArgs&quot;:{&quot;namespace&quot;:&quot;html&quot;,&quot;tag&quot;:&quot;div&quot;},&quot;children&quot;:[{&quot;props&quot;:{&quot;className&quot;:&quot;interact &quot;,&quot;style&quot;:{&quot;padding&quot;:&quot;5px 10px 0px 10px&quot;}},&quot;nodeType&quot;:&quot;DOM&quot;,&quot;type&quot;:&quot;node&quot;,&quot;instanceArgs&quot;:{&quot;namespace&quot;:&quot;html&quot;,&quot;tag&quot;:&quot;label&quot;},&quot;children&quot;:[&quot;&quot;]}]},{&quot;props&quot;:{&quot;attributes&quot;:{&quot;class&quot;:&quot;interact-flex-row-center&quot;}},&quot;nodeType&quot;:&quot;DOM&quot;,&quot;type&quot;:&quot;node&quot;,&quot;instanceArgs&quot;:{&quot;namespace&quot;:&quot;html&quot;,&quot;tag&quot;:&quot;div&quot;},&quot;children&quot;:[{&quot;props&quot;:{&quot;max&quot;:9,&quot;min&quot;:0,&quot;attributes&quot;:{&quot;type&quot;:&quot;range&quot;,&quot;data-bind&quot;:&quot;numericValue: value, valueUpdate: &#39;input&#39;, event: {change : function () {this.changes(this.changes()+1)}}&quot;,&quot;orient&quot;:&quot;horizontal&quot;},&quot;step&quot;:1,&quot;className&quot;:&quot;slider slider is-fullwidth&quot;,&quot;style&quot;:{}},&quot;nodeType&quot;:&quot;DOM&quot;,&quot;type&quot;:&quot;node&quot;,&quot;instanceArgs&quot;:{&quot;namespace&quot;:&quot;html&quot;,&quot;tag&quot;:&quot;input&quot;},&quot;children&quot;:[]}]},{&quot;props&quot;:{&quot;attributes&quot;:{&quot;class&quot;:&quot;interact-flex-row-right&quot;}},&quot;nodeType&quot;:&quot;DOM&quot;,&quot;type&quot;:&quot;node&quot;,&quot;instanceArgs&quot;:{&quot;namespace&quot;:&quot;html&quot;,&quot;tag&quot;:&quot;div&quot;},&quot;children&quot;:[{&quot;props&quot;:{&quot;attributes&quot;:{&quot;data-bind&quot;:&quot;text: value&quot;}},&quot;nodeType&quot;:&quot;DOM&quot;,&quot;type&quot;:&quot;node&quot;,&quot;instanceArgs&quot;:{&quot;namespace&quot;:&quot;html&quot;,&quot;tag&quot;:&quot;p&quot;},&quot;children&quot;:[]}]}]}]}]},{&quot;props&quot;:{&quot;className&quot;:&quot;field&quot;},&quot;nodeType&quot;:&quot;DOM&quot;,&quot;type&quot;:&quot;node&quot;,&quot;instanceArgs&quot;:{&quot;namespace&quot;:&quot;html&quot;,&quot;tag&quot;:&quot;div&quot;},&quot;children&quot;:[{&quot;props&quot;:{},&quot;nodeType&quot;:&quot;Scope&quot;,&quot;type&quot;:&quot;node&quot;,&quot;instanceArgs&quot;:{&quot;imports&quot;:{&quot;data&quot;:[{&quot;name&quot;:&quot;knockout&quot;,&quot;type&quot;:&quot;js&quot;,&quot;url&quot;:&quot;/assetserver/06c67466ebc6c52906ffd4101b7c74a47f9567bc-knockout.js&quot;},{&quot;name&quot;:&quot;knockout_punches&quot;,&quot;type&quot;:&quot;js&quot;,&quot;url&quot;:&quot;/assetserver/46b0ee2b2af2730ca29ddb46f091ac108fb8f609-knockout_punches.js&quot;},{&quot;name&quot;:null,&quot;type&quot;:&quot;js&quot;,&quot;url&quot;:&quot;/assetserver/e923a2cf7cc8bb183f572d980ae606c5f08fa0dc-all.js&quot;},{&quot;name&quot;:null,&quot;type&quot;:&quot;css&quot;,&quot;url&quot;:&quot;/assetserver/b973c333a7d1e5a6ca74f1442a2f26b5e4d0580e-style.css&quot;},{&quot;name&quot;:null,&quot;type&quot;:&quot;css&quot;,&quot;url&quot;:&quot;/assetserver/885f9c98e35e2ecc817c654a867b4570a606ec22-main.css&quot;}],&quot;type&quot;:&quot;async_block&quot;},&quot;id&quot;:&quot;knockout-component-79402efb-8e52-4a9b-8be2-461225e7e49c&quot;,&quot;handlers&quot;:{&quot;_promises&quot;:{&quot;importsLoaded&quot;:[function (ko, koPunches) {\n",
       "    ko.punches.enableAll();\n",
       "    ko.bindingHandlers.numericValue = {\n",
       "        init : function(element, valueAccessor, allBindings, data, context) {\n",
       "            var stringified = ko.observable(ko.unwrap(valueAccessor()));\n",
       "            stringified.subscribe(function(value) {\n",
       "                var val = parseFloat(value);\n",
       "                if (!isNaN(val)) {\n",
       "                    valueAccessor()(val);\n",
       "                }\n",
       "            })\n",
       "            valueAccessor().subscribe(function(value) {\n",
       "                var str = JSON.stringify(value);\n",
       "                if ((str == &quot;0&quot;) &amp;&amp; ([&quot;-0&quot;, &quot;-0.&quot;].indexOf(stringified()) &gt;= 0))\n",
       "                     return;\n",
       "                 if ([&quot;null&quot;, &quot;&quot;].indexOf(str) &gt;= 0)\n",
       "                     return;\n",
       "                stringified(str);\n",
       "            })\n",
       "            ko.applyBindingsToNode(element, { value: stringified, valueUpdate: allBindings.get(&#39;valueUpdate&#39;)}, context);\n",
       "        }\n",
       "    };\n",
       "    var json_data = JSON.parse(&quot;{\\&quot;changes\\&quot;:0,\\&quot;value\\&quot;:1}&quot;);\n",
       "    var self = this;\n",
       "    function AppViewModel() {\n",
       "        for (var key in json_data) {\n",
       "            var el = json_data[key];\n",
       "            this[key] = Array.isArray(el) ? ko.observableArray(el) : ko.observable(el);\n",
       "        }\n",
       "        \n",
       "        \n",
       "        [this[&quot;changes&quot;].subscribe((function (val){!(this.valueFromJulia[&quot;changes&quot;]) ? (WebIO.setval({&quot;name&quot;:&quot;changes&quot;,&quot;scope&quot;:&quot;knockout-component-79402efb-8e52-4a9b-8be2-461225e7e49c&quot;,&quot;id&quot;:&quot;ob_61&quot;,&quot;type&quot;:&quot;observable&quot;},val)) : undefined; return this.valueFromJulia[&quot;changes&quot;]=false}),self),this[&quot;value&quot;].subscribe((function (val){!(this.valueFromJulia[&quot;value&quot;]) ? (WebIO.setval({&quot;name&quot;:&quot;value&quot;,&quot;scope&quot;:&quot;knockout-component-79402efb-8e52-4a9b-8be2-461225e7e49c&quot;,&quot;id&quot;:&quot;ob_60&quot;,&quot;type&quot;:&quot;observable&quot;},val)) : undefined; return this.valueFromJulia[&quot;value&quot;]=false}),self)]\n",
       "        \n",
       "    }\n",
       "    self.model = new AppViewModel();\n",
       "    self.valueFromJulia = {};\n",
       "    for (var key in json_data) {\n",
       "        self.valueFromJulia[key] = false;\n",
       "    }\n",
       "    ko.applyBindings(self.model, self.dom);\n",
       "}\n",
       "]},&quot;changes&quot;:[(function (val){return (val!=this.model[&quot;changes&quot;]()) ? (this.valueFromJulia[&quot;changes&quot;]=true, this.model[&quot;changes&quot;](val)) : undefined})],&quot;value&quot;:[(function (val){return (val!=this.model[&quot;value&quot;]()) ? (this.valueFromJulia[&quot;value&quot;]=true, this.model[&quot;value&quot;](val)) : undefined})]},&quot;systemjs_options&quot;:null,&quot;observables&quot;:{&quot;changes&quot;:{&quot;sync&quot;:false,&quot;id&quot;:&quot;ob_61&quot;,&quot;value&quot;:0},&quot;value&quot;:{&quot;sync&quot;:true,&quot;id&quot;:&quot;ob_60&quot;,&quot;value&quot;:1}}},&quot;children&quot;:[{&quot;props&quot;:{&quot;attributes&quot;:{&quot;class&quot;:&quot;interact-flex-row&quot;}},&quot;nodeType&quot;:&quot;DOM&quot;,&quot;type&quot;:&quot;node&quot;,&quot;instanceArgs&quot;:{&quot;namespace&quot;:&quot;html&quot;,&quot;tag&quot;:&quot;div&quot;},&quot;children&quot;:[{&quot;props&quot;:{&quot;attributes&quot;:{&quot;class&quot;:&quot;interact-flex-row-left&quot;}},&quot;nodeType&quot;:&quot;DOM&quot;,&quot;type&quot;:&quot;node&quot;,&quot;instanceArgs&quot;:{&quot;namespace&quot;:&quot;html&quot;,&quot;tag&quot;:&quot;div&quot;},&quot;children&quot;:[{&quot;props&quot;:{&quot;className&quot;:&quot;interact &quot;,&quot;style&quot;:{&quot;padding&quot;:&quot;5px 10px 0px 10px&quot;}},&quot;nodeType&quot;:&quot;DOM&quot;,&quot;type&quot;:&quot;node&quot;,&quot;instanceArgs&quot;:{&quot;namespace&quot;:&quot;html&quot;,&quot;tag&quot;:&quot;label&quot;},&quot;children&quot;:[&quot;&quot;]}]},{&quot;props&quot;:{&quot;attributes&quot;:{&quot;class&quot;:&quot;interact-flex-row-center&quot;}},&quot;nodeType&quot;:&quot;DOM&quot;,&quot;type&quot;:&quot;node&quot;,&quot;instanceArgs&quot;:{&quot;namespace&quot;:&quot;html&quot;,&quot;tag&quot;:&quot;div&quot;},&quot;children&quot;:[{&quot;props&quot;:{&quot;max&quot;:10,&quot;min&quot;:1,&quot;attributes&quot;:{&quot;type&quot;:&quot;range&quot;,&quot;data-bind&quot;:&quot;numericValue: value, valueUpdate: &#39;input&#39;, event: {change : function () {this.changes(this.changes()+1)}}&quot;,&quot;orient&quot;:&quot;horizontal&quot;},&quot;step&quot;:1,&quot;className&quot;:&quot;slider slider is-fullwidth&quot;,&quot;style&quot;:{}},&quot;nodeType&quot;:&quot;DOM&quot;,&quot;type&quot;:&quot;node&quot;,&quot;instanceArgs&quot;:{&quot;namespace&quot;:&quot;html&quot;,&quot;tag&quot;:&quot;input&quot;},&quot;children&quot;:[]}]},{&quot;props&quot;:{&quot;attributes&quot;:{&quot;class&quot;:&quot;interact-flex-row-right&quot;}},&quot;nodeType&quot;:&quot;DOM&quot;,&quot;type&quot;:&quot;node&quot;,&quot;instanceArgs&quot;:{&quot;namespace&quot;:&quot;html&quot;,&quot;tag&quot;:&quot;div&quot;},&quot;children&quot;:[{&quot;props&quot;:{&quot;attributes&quot;:{&quot;data-bind&quot;:&quot;text: value&quot;}},&quot;nodeType&quot;:&quot;DOM&quot;,&quot;type&quot;:&quot;node&quot;,&quot;instanceArgs&quot;:{&quot;namespace&quot;:&quot;html&quot;,&quot;tag&quot;:&quot;p&quot;},&quot;children&quot;:[]}]}]}]}]},{&quot;props&quot;:{},&quot;nodeType&quot;:&quot;Scope&quot;,&quot;type&quot;:&quot;node&quot;,&quot;instanceArgs&quot;:{&quot;imports&quot;:{&quot;data&quot;:[],&quot;type&quot;:&quot;async_block&quot;},&quot;id&quot;:&quot;scope-be6da3ab-84c5-4a34-97fa-500526ca7424&quot;,&quot;handlers&quot;:{&quot;obs-output&quot;:[function (updated_htmlstr) {\n",
       "    var el = this.dom.querySelector(&quot;#out&quot;);\n",
       "    WebIO.propUtils.setInnerHtml(el, updated_htmlstr);\n",
       "}]},&quot;systemjs_options&quot;:null,&quot;observables&quot;:{&quot;obs-output&quot;:{&quot;sync&quot;:false,&quot;id&quot;:&quot;ob_64&quot;,&quot;value&quot;:&quot;&lt;div class=&#39;display:none&#39;&gt;&lt;/div&gt;&lt;unsafe-script style=&#39;display:none&#39;&gt;\\nWebIO.mount(this.previousSibling,{&amp;quot;props&amp;quot;:{&amp;quot;attributes&amp;quot;:{&amp;quot;class&amp;quot;:&amp;quot;interact-flex-row&amp;quot;}},&amp;quot;nodeType&amp;quot;:&amp;quot;DOM&amp;quot;,&amp;quot;type&amp;quot;:&amp;quot;node&amp;quot;,&amp;quot;instanceArgs&amp;quot;:{&amp;quot;namespace&amp;quot;:&amp;quot;html&amp;quot;,&amp;quot;tag&amp;quot;:&amp;quot;div&amp;quot;},&amp;quot;children&amp;quot;:[{&amp;quot;props&amp;quot;:{&amp;quot;setInnerHtml&amp;quot;:&amp;quot;&amp;lt;img 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class=&#39;display:none&#39;&gt;&lt;/div&gt;&lt;unsafe-script style=&#39;display:none&#39;&gt;\\nWebIO.mount(this.previousSibling,{&amp;quot;props&amp;quot;:{&amp;quot;attributes&amp;quot;:{&amp;quot;class&amp;quot;:&amp;quot;interact-flex-row&amp;quot;}},&amp;quot;nodeType&amp;quot;:&amp;quot;DOM&amp;quot;,&amp;quot;type&amp;quot;:&amp;quot;node&amp;quot;,&amp;quot;instanceArgs&amp;quot;:{&amp;quot;namespace&amp;quot;:&amp;quot;html&amp;quot;,&amp;quot;tag&amp;quot;:&amp;quot;div&amp;quot;},&amp;quot;children&amp;quot;:[{&amp;quot;props&amp;quot;:{&amp;quot;setInnerHtml&amp;quot;:&amp;quot;&amp;lt;img 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       "</div>"
      ],
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       "Widget{:manipulate,Any}(OrderedDict{Symbol,Any}(:digit=>Widget{:slider,Int64}(OrderedDict{Symbol,Any}(:changes=>Observable{Int64} with 1 listeners. Value:\n",
       "0,:value=>Observable{Int64} with 2 listeners. Value:\n",
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       "1), Observable{Int64} with 2 listeners. Value:\n",
       "1, Scope(\"knockout-component-79402efb-8e52-4a9b-8be2-461225e7e49c\", Node{DOM}(DOM(:html, :div), Any[Node{DOM}(DOM(:html, :div), Any[Node{DOM}(DOM(:html, :label), Any[nothing], Dict{Symbol,Any}(:className=>\"interact \",:style=>Dict{Any,Any}(:padding=>\"5px 10px 0px 10px\")), 1)], Dict{Symbol,Any}(:attributes=>Dict(\"class\"=>\"interact-flex-row-left\")), 2), Node{DOM}(DOM(:html, :div), Any[Node{DOM}(DOM(:html, :input), Any[], Dict{Symbol,Any}(:max=>10,:min=>1,:attributes=>Dict{Any,Any}(:type=>\"range\",Symbol(\"data-bind\")=>\"numericValue: value, valueUpdate: 'input', event: {change : function () {this.changes(this.changes()+1)}}\",\"orient\"=>\"horizontal\"),:step=>1,:className=>\"slider slider is-fullwidth\",:style=>Dict{Any,Any}()), 0)], Dict{Symbol,Any}(:attributes=>Dict(\"class\"=>\"interact-flex-row-center\")), 1), Node{DOM}(DOM(:html, :div), Any[Node{DOM}(DOM(:html, :p), Any[], Dict{Symbol,Any}(:attributes=>Dict(\"data-bind\"=>\"text: value\")), 0)], Dict{Symbol,Any}(:attributes=>Dict(\"class\"=>\"interact-flex-row-right\")), 1)], Dict{Symbol,Any}(:attributes=>Dict(\"class\"=>\"interact-flex-row\")), 7), Dict{String,Tuple{Observables.AbstractObservable,Union{Nothing, Bool}}}(\"changes\"=>(Observable{Int64} with 1 listeners. Value:\n",
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       "1, nothing)), Set(String[]), nothing, Any[\"knockout\"=>\"/Users/alanedelman/.julia/packages/Knockout/JIqpG/src/../assets/knockout.js\", \"knockout_punches\"=>\"/Users/alanedelman/.julia/packages/Knockout/JIqpG/src/../assets/knockout_punches.js\", \"/Users/alanedelman/.julia/packages/InteractBase/PTCUD/src/../assets/all.js\", \"/Users/alanedelman/.julia/packages/InteractBase/PTCUD/src/../assets/style.css\", \"/Users/alanedelman/.julia/packages/InteractBulma/PH56C/src/../assets/main.css\"], Dict{Any,Any}(\"_promises\"=>Dict{Any,Any}(\"importsLoaded\"=>Any[JSString(\"function (ko, koPunches) {\\n    ko.punches.enableAll();\\n    ko.bindingHandlers.numericValue = {\\n        init : function(element, valueAccessor, allBindings, data, context) {\\n            var stringified = ko.observable(ko.unwrap(valueAccessor()));\\n            stringified.subscribe(function(value) {\\n                var val = parseFloat(value);\\n                if (!isNaN(val)) {\\n                    valueAccessor()(val);\\n                }\\n            })\\n            valueAccessor().subscribe(function(value) {\\n                var str = JSON.stringify(value);\\n                if ((str == \\\"0\\\") && ([\\\"-0\\\", \\\"-0.\\\"].indexOf(stringified()) >= 0))\\n                     return;\\n                 if ([\\\"null\\\", \\\"\\\"].indexOf(str) >= 0)\\n                     return;\\n                stringified(str);\\n            })\\n            ko.applyBindingsToNode(element, { value: stringified, valueUpdate: allBindings.get('valueUpdate')}, context);\\n        }\\n    };\\n    var json_data = JSON.parse(\\\"{\\\\\\\"changes\\\\\\\":0,\\\\\\\"value\\\\\\\":1}\\\");\\n    var self = this;\\n    function AppViewModel() {\\n        for (var key in json_data) {\\n            var el = json_data[key];\\n            this[key] = Array.isArray(el) ? ko.observableArray(el) : ko.observable(el);\\n        }\\n        \\n        \\n        [this[\\\"changes\\\"].subscribe((function (val){!(this.valueFromJulia[\\\"changes\\\"]) ? (WebIO.setval({\\\"name\\\":\\\"changes\\\",\\\"scope\\\":\\\"knockout-component-79402efb-8e52-4a9b-8be2-461225e7e49c\\\",\\\"id\\\":\\\"ob_61\\\",\\\"type\\\":\\\"observable\\\"},val)) : undefined; return this.valueFromJulia[\\\"changes\\\"]=false}),self),this[\\\"value\\\"].subscribe((function (val){!(this.valueFromJulia[\\\"value\\\"]) ? (WebIO.setval({\\\"name\\\":\\\"value\\\",\\\"scope\\\":\\\"knockout-component-79402efb-8e52-4a9b-8be2-461225e7e49c\\\",\\\"id\\\":\\\"ob_60\\\",\\\"type\\\":\\\"observable\\\"},val)) : undefined; return this.valueFromJulia[\\\"value\\\"]=false}),self)]\\n        \\n    }\\n    self.model = new AppViewModel();\\n    self.valueFromJulia = {};\\n    for (var key in json_data) {\\n        self.valueFromJulia[key] = false;\\n    }\\n    ko.applyBindings(self.model, self.dom);\\n}\\n\")]),\"changes\"=>Any[JSString(\"(function (val){return (val!=this.model[\\\"changes\\\"]()) ? (this.valueFromJulia[\\\"changes\\\"]=true, this.model[\\\"changes\\\"](val)) : undefined})\")],\"value\"=>Any[JSString(\"(function (val){return (val!=this.model[\\\"value\\\"]()) ? (this.valueFromJulia[\\\"value\\\"]=true, this.model[\\\"value\\\"](val)) : undefined})\")]), ConnectionPool(Channel{Any}(sz_max:9223372036854775807,sz_curr:3), Set(AbstractConnection[]), Channel{AbstractConnection}(sz_max:32,sz_curr:0))), ##52#53{#dom#15{##dom#13#14{Dict{Any,Any},DOM}},typeof(scope)}(#dom#15{##dom#13#14{Dict{Any,Any},DOM}}(##dom#13#14{Dict{Any,Any},DOM}(Dict{Any,Any}(:className=>\"field\"), DOM(:html, :div))), scope))), Observable{Any} with 0 listeners. Value:\n",
       "Gray{Float64}[Gray{Float64}(1.19636e-15) Gray{Float64}(0.0) … Gray{Float64}(0.0) Gray{Float64}(0.0); Gray{Float64}(0.0) Gray{Float64}(0.0) … Gray{Float64}(0.0) Gray{Float64}(0.0); … ; Gray{Float64}(0.0) Gray{Float64}(0.0) … Gray{Float64}(0.0) Gray{Float64}(0.0); Gray{Float64}(0.0) Gray{Float64}(0.0) … Gray{Float64}(0.0) Gray{Float64}(0.0)], nothing, getfield(InteractBase, Symbol(\"##142#143\"))())"
      ]
     },
     "execution_count": 74,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "@manipulate for digit in slider(0:9, value=0), which_sing_vec in slider(1:10, value=1)\n",
    "\n",
    "    image = Us[digit+1][:,which_sing_vec]    \n",
    "   \n",
    "    normalize_image(image)\n",
    "end"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Predicting labels from images of digits\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's use the SVD to try to **predict** a label for an image that we haven't seen yet:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 75,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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",
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       " Gray{N0f8}(0.0)  Gray{N0f8}(0.0)     Gray{N0f8}(0.0)  Gray{N0f8}(0.0)\n",
       " Gray{N0f8}(0.0)  Gray{N0f8}(0.0)     Gray{N0f8}(0.0)  Gray{N0f8}(0.0)\n",
       " ⋮                                 ⋱                                  \n",
       " Gray{N0f8}(0.0)  Gray{N0f8}(0.0)     Gray{N0f8}(0.0)  Gray{N0f8}(0.0)\n",
       " Gray{N0f8}(0.0)  Gray{N0f8}(0.0)     Gray{N0f8}(0.0)  Gray{N0f8}(0.0)\n",
       " Gray{N0f8}(0.0)  Gray{N0f8}(0.0)     Gray{N0f8}(0.0)  Gray{N0f8}(0.0)\n",
       " Gray{N0f8}(0.0)  Gray{N0f8}(0.0)     Gray{N0f8}(0.0)  Gray{N0f8}(0.0)\n",
       " Gray{N0f8}(0.0)  Gray{N0f8}(0.0)  …  Gray{N0f8}(0.0)  Gray{N0f8}(0.0)\n",
       " Gray{N0f8}(0.0)  Gray{N0f8}(0.0)     Gray{N0f8}(0.0)  Gray{N0f8}(0.0)\n",
       " Gray{N0f8}(0.0)  Gray{N0f8}(0.0)     Gray{N0f8}(0.0)  Gray{N0f8}(0.0)\n",
       " Gray{N0f8}(0.0)  Gray{N0f8}(0.0)     Gray{N0f8}(0.0)  Gray{N0f8}(0.0)\n",
       " Gray{N0f8}(0.0)  Gray{N0f8}(0.0)     Gray{N0f8}(0.0)  Gray{N0f8}(0.0)\n",
       " Gray{N0f8}(0.0)  Gray{N0f8}(0.0)  …  Gray{N0f8}(0.0)  Gray{N0f8}(0.0)\n",
       " Gray{N0f8}(0.0)  Gray{N0f8}(0.0)     Gray{N0f8}(0.0)  Gray{N0f8}(0.0)\n",
       " Gray{N0f8}(0.0)  Gray{N0f8}(0.0)     Gray{N0f8}(0.0)  Gray{N0f8}(0.0)"
      ]
     },
     "execution_count": 75,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "image = images[training_size + 1]   # was not included in the training"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 88,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "10-element Array{Array{Float64,2},1}:\n",
       " [-1.11022e-16 -8.32667e-17 -1.80411e-16; 0.0 1.11022e-16 -2.77556e-17; … ; 0.0 0.0 0.0; 0.0 0.0 0.0]\n",
       " [3.33067e-16 2.22045e-16 -1.38778e-16; 0.0 1.66533e-16 1.11022e-16; … ; 0.0 0.0 0.0; 0.0 0.0 0.0]   \n",
       " [2.22045e-16 -5.55112e-17 5.55112e-17; 0.0 1.94289e-16 -5.55112e-17; … ; 0.0 0.0 0.0; 0.0 0.0 0.0]  \n",
       " [-1.11022e-16 -5.55112e-17 5.55112e-17; 0.0 -2.77556e-16 2.22045e-16; … ; 0.0 0.0 0.0; 0.0 0.0 0.0] \n",
       " [1.11022e-16 0.0 1.94289e-16; 0.0 1.249e-16 -1.66533e-16; … ; 0.0 0.0 0.0; 0.0 0.0 0.0]             \n",
       " [-3.33067e-16 -1.38778e-17 5.55112e-17; 0.0 -8.67362e-18 -1.94289e-16; … ; 0.0 0.0 0.0; 0.0 0.0 0.0]\n",
       " [-1.11022e-16 -6.245e-17 -6.93889e-17; 0.0 3.46945e-17 -1.94289e-16; … ; 0.0 0.0 0.0; 0.0 0.0 0.0]  \n",
       " [0.0 1.66533e-16 0.0; 0.0 -1.66533e-16 0.0; … ; 0.0 0.0 0.0; 0.0 0.0 0.0]                           \n",
       " [2.22045e-16 5.55112e-17 -8.32667e-17; 0.0 -2.77556e-16 2.498e-16; … ; 0.0 0.0 0.0; 0.0 0.0 0.0]    \n",
       " [1.11022e-16 -1.94289e-16 1.38778e-16; 0.0 2.22045e-16 -8.32667e-17; … ; 0.0 0.0 0.0; 0.0 0.0 0.0]  "
      ]
     },
     "execution_count": 88,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "M = [U[:,1:3] for U in Us]  # first n sing vecs for each digit"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 89,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "784×10 Array{Float64,2}:\n",
       " -1.11022e-16   3.33067e-16   2.22045e-16  …  2.22045e-16  1.11022e-16\n",
       "  0.0           0.0           0.0             0.0          0.0        \n",
       "  0.0           0.0           0.0             0.0          0.0        \n",
       "  0.0           0.0           0.0             0.0          0.0        \n",
       "  0.0           0.0           0.0             0.0          0.0        \n",
       "  0.0           0.0           0.0          …  0.0          0.0        \n",
       "  0.0           0.0           0.0             0.0          0.0        \n",
       "  0.0           0.0           0.0             0.0          0.0        \n",
       "  0.0           0.0           0.0             0.0          0.0        \n",
       "  0.0           0.0           0.0             0.0          0.0        \n",
       "  0.0           0.0           0.0          …  0.0          0.0        \n",
       "  0.0           0.0           0.0             0.0          0.0        \n",
       "  0.0           0.0           0.0             0.0          0.0        \n",
       "  ⋮                                        ⋱                          \n",
       " -4.4301e-7     0.0          -3.147e-5        0.0          0.0        \n",
       " -3.83942e-6   -8.10781e-7   -2.42268e-5      0.0          0.0        \n",
       "  0.0          -3.49319e-5   -3.07186e-5      0.0          0.0        \n",
       "  0.0          -1.94591e-5   -1.85176e-5   …  0.0          0.0        \n",
       "  0.0           0.0           0.0             0.0          0.0        \n",
       " -3.91902e-6    0.0           0.0             0.0          0.0        \n",
       " -7.23511e-6    0.0           0.0             0.0          0.0        \n",
       "  0.0           0.0           0.0             0.0          0.0        \n",
       "  0.0           0.0           0.0          …  0.0          0.0        \n",
       "  0.0           0.0           0.0             0.0          0.0        \n",
       "  0.0           0.0           0.0             0.0          0.0        \n",
       "  0.0           0.0           0.0             0.0          0.0        "
      ]
     },
     "execution_count": 89,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "first_sing_vecs = reduce(hcat, [U[:,1] for U in Us])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 90,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "my_softmax (generic function with 1 method)"
      ]
     },
     "execution_count": 90,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "my_softmax(v) = exp.(v) / sum(exp.(v))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 91,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "10-element Array{Float64,1}:\n",
       " 0.03382072710940378 \n",
       " 0.15392804472033755 \n",
       " 0.12316884381500334 \n",
       " 0.2824501000116202  \n",
       " 0.027879155990251033\n",
       " 0.12441004437229447 \n",
       " 0.0474266712208733  \n",
       " 0.02423256810297591 \n",
       " 0.1465310863057624  \n",
       " 0.0361527583514779  "
      ]
     },
     "execution_count": 91,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "v = abs.(first_sing_vecs' * vec(Float64.(image)))\n",
    "\n",
    "my_softmax(v)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 92,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "10-element Array{Float64,1}:\n",
       " 0.033820727109403786\n",
       " 0.15392804472033758 \n",
       " 0.12316884381500348 \n",
       " 0.28245010001161996 \n",
       " 0.027879155990251016\n",
       " 0.1244100443722945  \n",
       " 0.04742667122087331 \n",
       " 0.024232568102975912\n",
       " 0.14653108630576242 \n",
       " 0.03615275835147791 "
      ]
     },
     "execution_count": 92,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "M = [U[:, 1:1] for U in Us]  # first n sing vecs for each digit\n",
    "\n",
    "my_softmax(norm.([MM' * vec(Float64.(image)) for MM in M]))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 93,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
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     "execution_count": 93,
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   "source": [
    "@manipulate for num_sing_vecs in 1:784\n",
    "\n",
    "    M = [U[:, 1:num_sing_vecs] for U in Us]  # first n sing vecs for each digit\n",
    "\n",
    "    v = my_softmax(norm.([MM' * vec(Float64.(image)) for MM in M]))\n",
    "    \n",
    "    hbox(v, plot(0:9, v, m=:o), findmax(v))\n",
    "    \n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 96,
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     "execution_count": 96,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "data = []\n",
    "\n",
    "for num_sing_vecs in 1:15\n",
    "\n",
    "    M = [U[:, 1:num_sing_vecs] for U in Us]  # first n sing vecs for each digit\n",
    "\n",
    "    v = my_softmax(norm.([MM' * vec(Float64.(image)) for MM in M]))\n",
    "    \n",
    "    push!(data, v[4])\n",
    "end\n",
    "\n",
    "plot(data, m=:o)\n",
    "scatter!([argmax(data)], [maximum(data)])\n",
    "    "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 97,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "6"
      ]
     },
     "execution_count": 97,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "argmax(data)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 98,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "10-element Array{Float64,1}:\n",
       " 0.06244873229934407 \n",
       " 0.08153335605992665 \n",
       " 0.14844813863333253 \n",
       " 0.3010347736526581  \n",
       " 0.0248050897247611  \n",
       " 0.11181027499468853 \n",
       " 0.06919557999052471 \n",
       " 0.025800589190491042\n",
       " 0.13536138735349518 \n",
       " 0.03956207810077807 "
      ]
     },
     "execution_count": 98,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "softmax(norm.([MM' * vec(Float64.(image)) for MM in M]))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 99,
   "metadata": {},
   "outputs": [
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