{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "# Euclidean Projection onto Unit Balls and Spheres\n", "\n", "In this notebook we provide examples of projecting points onto unit $L_p$ [balls and spheres](https://en.wikipedia.org/wiki/Ball_%28mathematics%29) by solving the following constrained optimization problems\n", "\n", "$$\n", "\\begin{array}{lll}\n", " y \\in & \\text{argmin}_u & \\frac{1}{2} \\|u - x\\|^2 \\\\\n", " & \\text{subject to} & \\|u\\|_p \\leq 1\n", "\\end{array}\n", "$$\n", "and\n", "$$\n", "\\begin{array}{lll}\n", " y \\in & \\text{argmin}_u & \\frac{1}{2} \\|u - x\\|^2 \\\\\n", " & \\text{subject to} & \\|u\\|_p = 1\n", "\\end{array}\n", "$$\n", "respectively. We consider three balls $p = 1, 2$ and $\\infty$.\n" ] }, { "cell_type": "code", "execution_count": 1, "metadata": {}, "outputs": [], "source": [ "import numpy as np\n", "\n", "\n", "def insideBall(x, p=2):\n", " \"\"\"Returns true if point x is inside a unit Lp ball.\"\"\"\n", " return np.linalg.norm(x, p) <= 1.0\n", "\n", "\n", "def euclideanProjection(x, p=2):\n", " \"\"\"Projects the point x onto the unit Lp sphere.\"\"\"\n", " if (p == 1):\n", " mu = np.sort(np.abs(x))[::-1]\n", " mu_sum = mu.cumsum()\n", " rho = np.max([j for j in range(len(mu)) if ((j + 1) * mu[j] > mu_sum[j] - 1.0)])\n", " theta = (mu_sum[rho] - 1.0) / (rho + 1)\n", " w = np.maximum(np.abs(x) - theta, 0.0)\n", " return w * [1.0 if xi >= 0 else -1.0 for xi in x]\n", " if (p == 2):\n", " return x / np.sqrt(np.dot(x, x))\n", " if (p == np.inf):\n", " w = np.minimum(np.maximum(x, -1.0), 1.0)\n", " if np.all(np.abs(w) < 1.0):\n", " rho = np.argmin(1.0 - np.abs(w))\n", " w[rho] = np.sign(w[rho])\n", " return w\n", " \n", " assert False, \"unrecognized p={}\".format(p)\n", " return None\n" ] }, { "cell_type": "markdown", "metadata": { "collapsed": true }, "source": [ "We now animate projection onto different unit spheres as a function of $x$ for the 2-dimensional point $(x, 0.5)$ shown as the black dot. The red, blue and green dots represent the nearest points to $(x, 0.5)$ on the $L_1$-, $L_2$- and $L_\\infty$-spheres, respectively." ] }, { "cell_type": "code", "execution_count": 2, "metadata": {}, "outputs": [], "source": [ "%%capture\n", "%matplotlib notebook\n", "import matplotlib.pyplot as plt\n", "from matplotlib import animation\n", "from IPython.display import HTML\n", "\n", "\n", "x = np.linspace(-3.0, 3.0, num=101)\n", "y_1, y_2, y_inf = [], [], []\n", "\n", "for xi in x:\n", " y_1.append(euclideanProjection([xi, 0.5], 1))\n", " y_2.append(euclideanProjection([xi, 0.5], 2))\n", " y_inf.append(euclideanProjection([xi, 0.5], np.inf))\n", "\n", " \n", "def animate(fnum, x, y_1, y_2, y_inf):\n", "\n", " # draw balls\n", " ax.clear()\n", " ax.plot([0.0, 1.0, 0.0, -1.0, 0.0], [1.0, 0.0, -1.0, 0.0, 1.0], 'r', linewidth=1)\n", " ax.plot(np.cos(np.linspace(0.0, 2.0 * np.pi)), np.sin(np.linspace(0.0, 2.0 * np.pi)), 'b', linewidth=1)\n", " ax.plot([-1.0, 1.0, 1.0, -1.0, -1.0], [1.0, 1.0, -1.0, -1.0, 1.0], 'g', linewidth=1)\n", " \n", " # draw path for x\n", " ax.plot([x[0], x[-1]], [0.5, 0.5], 'k--')\n", " ax.plot(x[fnum], 0.5, 'ko')\n", " \n", " # draw projection\n", " ax.plot(y_1[fnum][0], y_1[fnum][1], 'ro')\n", " ax.plot(y_2[fnum][0], y_2[fnum][1], 'bo')\n", " ax.plot(y_inf[fnum][0], y_inf[fnum][1], 'go')\n", " \n", " ax.set_xlim(x[0], x[-1]); ax.set_ylim(x[0], x[-1])\n", " ax.axis('equal')\n", " return (ax,)\n", "\n", "\n", "fig = plt.figure()\n", "ax = plt.subplot(1, 1, 1)\n", "ani = animation.FuncAnimation(fig, animate, init_func=None, fargs=(x, y_1, y_2, y_inf),\n", " interval=100, frames=len(x), blit=False, repeat=False)\n", "plt.close(fig)" ] }, { "cell_type": "code", "execution_count": 3, "metadata": {}, "outputs": [ { "data": { "text/html": [ "" ], "text/plain": [ "" ] }, "execution_count": 3, "metadata": {}, "output_type": "execute_result" } ], "source": [ "#HTML(ani.to_jshtml())\n", "HTML(ani.to_html5_video())" ] }, { "cell_type": "code", "execution_count": 4, "metadata": {}, "outputs": [ { "data": { "application/javascript": [ "/* Put everything inside the global mpl namespace */\n", "window.mpl = {};\n", "\n", "\n", "mpl.get_websocket_type = function() {\n", " if (typeof(WebSocket) !== 'undefined') {\n", " return WebSocket;\n", " } else if (typeof(MozWebSocket) !== 'undefined') {\n", " return MozWebSocket;\n", " } else {\n", " alert('Your browser does not have WebSocket support. ' +\n", " 'Please try Chrome, Safari or Firefox ≥ 6. ' +\n", " 'Firefox 4 and 5 are also supported but you ' +\n", " 'have to enable WebSockets in about:config.');\n", " };\n", "}\n", "\n", "mpl.figure = function(figure_id, websocket, ondownload, parent_element) {\n", " this.id = figure_id;\n", "\n", " this.ws = websocket;\n", "\n", " this.supports_binary = (this.ws.binaryType != undefined);\n", "\n", " if (!this.supports_binary) {\n", " var warnings = document.getElementById(\"mpl-warnings\");\n", " if (warnings) {\n", " warnings.style.display = 'block';\n", " warnings.textContent = (\n", " \"This browser does not support binary websocket messages. \" +\n", " \"Performance may be slow.\");\n", " }\n", " }\n", "\n", " this.imageObj = new Image();\n", "\n", " this.context = undefined;\n", " this.message = undefined;\n", " this.canvas = undefined;\n", " this.rubberband_canvas = undefined;\n", " this.rubberband_context = undefined;\n", " this.format_dropdown = undefined;\n", "\n", " this.image_mode = 'full';\n", "\n", " this.root = $(' ');\n", " this._root_extra_style(this.root)\n", " this.root.attr('style', 'display: inline-block');\n", "\n", "$(parent_element).append(this.root);\n", "\n", " this._init_header(this);\n", " this._init_canvas(this);\n", " this._init_toolbar(this);\n", "\n", " var fig = this;\n", "\n", " this.waiting = false;\n", "\n", " this.ws.onopen = function () {\n", " fig.send_message(\"supports_binary\", {value: fig.supports_binary});\n", " fig.send_message(\"send_image_mode\", {});\n", " if (mpl.ratio != 1) {\n", " fig.send_message(\"set_dpi_ratio\", {'dpi_ratio': mpl.ratio});\n", " }\n", " fig.send_message(\"refresh\", {});\n", " }\n", "\n", " this.imageObj.onload = function() {\n", " if (fig.image_mode == 'full') {\n", " // Full images could contain transparency (where diff images\n", " // almost always do), so we need to clear the canvas so that\n", " // there is no ghosting.\n", " fig.context.clearRect(0, 0, fig.canvas.width, fig.canvas.height);\n", " }\n", " fig.context.drawImage(fig.imageObj, 0, 0);\n", " };\n", "\n", " this.imageObj.onunload = function() {\n", " fig.ws.close();\n", " }\n", "\n", " this.ws.onmessage = this._make_on_message_function(this);\n", "\n", " this.ondownload = ondownload;\n", "}\n", "\n", "mpl.figure.prototype._init_header = function() {\n", " var titlebar = $(\n", " ' ');\n", " var titletext =$(\n", " '
');\n", " titlebar.append(titletext)\n", " this.root.append(titlebar);\n", " this.header = titletext[0];\n", "}\n", "\n", "\n", "\n", "mpl.figure.prototype._canvas_extra_style = function(canvas_div) {\n", "\n", "}\n", "\n", "\n", "mpl.figure.prototype._root_extra_style = function(canvas_div) {\n", "\n", "}\n", "\n", "mpl.figure.prototype._init_canvas = function() {\n", " var fig = this;\n", "\n", " var canvas_div = $(' ');\n", "\n", " canvas_div.attr('style', 'position: relative; clear: both; outline: 0');\n", "\n", " function canvas_keyboard_event(event) {\n", " return fig.key_event(event, event['data']);\n", " }\n", "\n", " canvas_div.keydown('key_press', canvas_keyboard_event);\n", " canvas_div.keyup('key_release', canvas_keyboard_event);\n", " this.canvas_div = canvas_div\n", " this._canvas_extra_style(canvas_div)\n", " this.root.append(canvas_div);\n", "\n", " var canvas =$('');\n", " canvas.addClass('mpl-canvas');\n", " canvas.attr('style', \"left: 0; top: 0; z-index: 0; outline: 0\")\n", "\n", " this.canvas = canvas[0];\n", " this.context = canvas[0].getContext(\"2d\");\n", "\n", " var backingStore = this.context.backingStorePixelRatio ||\n", "\tthis.context.webkitBackingStorePixelRatio ||\n", "\tthis.context.mozBackingStorePixelRatio ||\n", "\tthis.context.msBackingStorePixelRatio ||\n", "\tthis.context.oBackingStorePixelRatio ||\n", "\tthis.context.backingStorePixelRatio || 1;\n", "\n", " mpl.ratio = (window.devicePixelRatio || 1) / backingStore;\n", "\n", " var rubberband = $('');\n", " rubberband.attr('style', \"position: absolute; left: 0; top: 0; z-index: 1;\")\n", "\n", " var pass_mouse_events = true;\n", "\n", " canvas_div.resizable({\n", " start: function(event, ui) {\n", " pass_mouse_events = false;\n", " },\n", " resize: function(event, ui) {\n", " fig.request_resize(ui.size.width, ui.size.height);\n", " },\n", " stop: function(event, ui) {\n", " pass_mouse_events = true;\n", " fig.request_resize(ui.size.width, ui.size.height);\n", " },\n", " });\n", "\n", " function mouse_event_fn(event) {\n", " if (pass_mouse_events)\n", " return fig.mouse_event(event, event['data']);\n", " }\n", "\n", " rubberband.mousedown('button_press', mouse_event_fn);\n", " rubberband.mouseup('button_release', mouse_event_fn);\n", " // Throttle sequential mouse events to 1 every 20ms.\n", " rubberband.mousemove('motion_notify', mouse_event_fn);\n", "\n", " rubberband.mouseenter('figure_enter', mouse_event_fn);\n", " rubberband.mouseleave('figure_leave', mouse_event_fn);\n", "\n", " canvas_div.on(\"wheel\", function (event) {\n", " event = event.originalEvent;\n", " event['data'] = 'scroll'\n", " if (event.deltaY < 0) {\n", " event.step = 1;\n", " } else {\n", " event.step = -1;\n", " }\n", " mouse_event_fn(event);\n", " });\n", "\n", " canvas_div.append(canvas);\n", " canvas_div.append(rubberband);\n", "\n", " this.rubberband = rubberband;\n", " this.rubberband_canvas = rubberband[0];\n", " this.rubberband_context = rubberband[0].getContext(\"2d\");\n", " this.rubberband_context.strokeStyle = \"#000000\";\n", "\n", " this._resize_canvas = function(width, height) {\n", " // Keep the size of the canvas, canvas container, and rubber band\n", " // canvas in synch.\n", " canvas_div.css('width', width)\n", " canvas_div.css('height', height)\n", "\n", " canvas.attr('width', width * mpl.ratio);\n", " canvas.attr('height', height * mpl.ratio);\n", " canvas.attr('style', 'width: ' + width + 'px; height: ' + height + 'px;');\n", "\n", " rubberband.attr('width', width);\n", " rubberband.attr('height', height);\n", " }\n", "\n", " // Set the figure to an initial 600x600px, this will subsequently be updated\n", " // upon first draw.\n", " this._resize_canvas(600, 600);\n", "\n", " // Disable right mouse context menu.\n", "$(this.rubberband_canvas).bind(\"contextmenu\",function(e){\n", " return false;\n", " });\n", "\n", " function set_focus () {\n", " canvas.focus();\n", " canvas_div.focus();\n", " }\n", "\n", " window.setTimeout(set_focus, 100);\n", "}\n", "\n", "mpl.figure.prototype._init_toolbar = function() {\n", " var fig = this;\n", "\n", " var nav_element = $(' ');\n", " nav_element.attr('style', 'width: 100%');\n", " this.root.append(nav_element);\n", "\n", " // Define a callback function for later on.\n", " function toolbar_event(event) {\n", " return fig.toolbar_button_onclick(event['data']);\n", " }\n", " function toolbar_mouse_event(event) {\n", " return fig.toolbar_button_onmouseover(event['data']);\n", " }\n", "\n", " for(var toolbar_ind in mpl.toolbar_items) {\n", " var name = mpl.toolbar_items[toolbar_ind][0];\n", " var tooltip = mpl.toolbar_items[toolbar_ind][1];\n", " var image = mpl.toolbar_items[toolbar_ind][2];\n", " var method_name = mpl.toolbar_items[toolbar_ind][3];\n", "\n", " if (!name) {\n", " // put a spacer in here.\n", " continue;\n", " }\n", " var button =$('