{
"cells": [
{
"cell_type": "markdown",
"metadata": {},
"source": [
"* This is a python jupyter notebook on the persisten homology with a python-binding package, dionysus. (incomplete version at this stage)\n",
" + About dionysus : https://github.com/mrzv/dionysus\n",
" + Version : 0.0\n",
" + Last update : May 5, 2018\n",
"\n",
"* This is based on the tutorials in http://mrzv.org/software/dionysus2/\n",
"\n",
"I'm planning to write down some documenations on the persistent cohomology for beginners...
\n",
"(either in Japanese or in English, which I've not decieded yet)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"* You need the Boost C++ library\n",
" + If you use Ubuntu, only you need to do is\n",
" > sudo apt-get install libboost-all-dev\n",
"* If you get an error message like \"version `GLIBCXX_3.4.20' not found\", you should update libgcc package\n",
" > conda update libgcc\n",
"* If you still have troubles, you may ask Google teacher."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# 1. Beginning"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Let's try : import dionysus"
]
},
{
"cell_type": "code",
"execution_count": 2,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/plain": [
"'2.0.5'"
]
},
"execution_count": 2,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"import dionysus\n",
"dionysus.__version__"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### auxiliary functions to see the properties of objects"
]
},
{
"cell_type": "code",
"execution_count": 83,
"metadata": {
"collapsed": true
},
"outputs": [],
"source": [
"def dir_(x):\n",
" return [ s for s in dir(x) if s[0] != '_']\n",
"def dir__(x):\n",
" return [ (s, 'method') if callable(getattr(x, s)) else (s, 'non-method') for s in dir_(x) ]\n",
"def info(x, show_value=True):\n",
" print(type(x))\n",
" print(dir__(x))\n",
" if show_value:\n",
" print(x)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## - Simplex"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### define a simplex\n",
"* any data point is labeled by a integer\n",
" + Hereafter we call it label of the vertex"
]
},
{
"cell_type": "code",
"execution_count": 4,
"metadata": {
"collapsed": true
},
"outputs": [],
"source": [
"from dionysus import Simplex"
]
},
{
"cell_type": "code",
"execution_count": 110,
"metadata": {
"collapsed": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"\n",
"[('boundary', 'method'), ('data', 'non-method'), ('dimension', 'method'), ('join', 'method')]\n"
]
}
],
"source": [
"# Let's define a 3-simplex from 4-points (0,1,2,3)\n",
"n = 3\n",
"s = Simplex(range(n+1))\n",
"info(s, show_value=False)"
]
},
{
"cell_type": "code",
"execution_count": 73,
"metadata": {
"collapsed": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"4.0\n",
"3.0\n"
]
}
],
"source": [
"# We can associate a real value to the simplex, later, which we can interpret as the radius at which it appears in the nerve\n",
"print(s.data)\n",
"s.data = 3.\n",
"print(s.data)"
]
},
{
"cell_type": "code",
"execution_count": 99,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/plain": [
"<0,1,1,2,2> 100"
]
},
"execution_count": 99,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"# Remark : we can also define the degenerate simplices\n",
"Simplex([0,1,2,1,2], 100.)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### get the label of the vertices ( data points ) of the simplex"
]
},
{
"cell_type": "code",
"execution_count": 87,
"metadata": {
"collapsed": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"0\n",
"1\n",
"2\n",
"3\n",
"\n"
]
}
],
"source": [
"for v in s:\n",
" print(v)\n",
"print(type(v))"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### get the boudary facet of the simplex"
]
},
{
"cell_type": "code",
"execution_count": 90,
"metadata": {
"collapsed": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"dim. = 3\n",
"No. 0 : <1,2,3> 0\n",
"No. 1 : <0,2,3> 0\n",
"No. 2 : <0,1,3> 0\n",
"No. 3 : <0,1,2> 0\n",
"---\n",
"\n",
"[('boundary', 'method'), ('data', 'non-method'), ('dimension', 'method'), ('join', 'method')]\n",
"<0,1,2> 0\n"
]
}
],
"source": [
"print(\"dim. = \", s.dimension())\n",
"for i, v in enumerate(s.boundary()):\n",
" print(\"No. {0} : {1}\".format(i, v))# i-th (n-1)-simplex, its value associated to the i-th boudary face\n",
"print('---')\n",
"info(v)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### generate all faces whose dimensions are lower than a given value"
]
},
{
"cell_type": "code",
"execution_count": 55,
"metadata": {
"collapsed": true
},
"outputs": [],
"source": [
"from dionysus import closure"
]
},
{
"cell_type": "code",
"execution_count": 93,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/plain": [
"[<0,2> 0,\n",
" <0,1> 0,\n",
" <1,3> 0,\n",
" <0,3> 0,\n",
" <1,2> 0,\n",
" <2,3> 0,\n",
" <0> 0,\n",
" <1> 0,\n",
" <2> 0,\n",
" <3> 0]"
]
},
"execution_count": 93,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"closure([s], 1)# all the 1-faces and 0-faces"
]
},
{
"cell_type": "code",
"execution_count": 92,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/plain": [
"[<0,2,3> 0,\n",
" <0,1,3> 0,\n",
" <0,2> 0,\n",
" <0,1> 0,\n",
" <1,3> 0,\n",
" <0,3> 0,\n",
" <1,2> 0,\n",
" <2,3> 0,\n",
" <1,2,3> 0,\n",
" <0> 0,\n",
" <1> 0,\n",
" <2> 0,\n",
" <3> 0,\n",
" <0,1,2> 0]"
]
},
"execution_count": 92,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"closure([s], 2)# all the 2-faces, the 1-faces and 0-faces"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## - Filtration ::: under construction"
]
},
{
"cell_type": "code",
"execution_count": 100,
"metadata": {
"collapsed": true
},
"outputs": [],
"source": [
"from dionysus import Filtration"
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {
"collapsed": true
},
"outputs": [],
"source": []
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {
"collapsed": true
},
"outputs": [],
"source": []
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## - Vietoris-Rips complexes"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### generate data points and plot them"
]
},
{
"cell_type": "code",
"execution_count": 215,
"metadata": {
"collapsed": true
},
"outputs": [],
"source": [
"import numpy as np\n",
"\n",
"num_data_points = 20\n",
"dim_feature_space = 2\n",
"\n",
"np.random.seed(10)\n",
"X = np.random.rand(num_data_points, dim_feature_space)"
]
},
{
"cell_type": "code",
"execution_count": 102,
"metadata": {
"collapsed": true
},
"outputs": [],
"source": [
"import matplotlib.pyplot as plt\n",
"%matplotlib inline\n",
"import matplotlib.cm as cm"
]
},
{
"cell_type": "code",
"execution_count": 216,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
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jY6OmYkgorVixgpaWlkFJORKJMHfu3ICiyjKjBm74Izh4AHp7Yc5cqFSJbzLLZ0TdDCzM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\n4LckjDwAAAAASUVORK5CYII=\n",
"text/plain": [
""
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"plt.xlim([-0.05, 1.05])\n",
"plt.ylim([-0.05, 1.05])\n",
"plt.scatter(X[:,0], X[:,1], c=np.arange(len(X)), cmap=cm.tab20)\n",
"plt.colorbar();"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Get the filtration of the VR complex as the sequence of the simplices"
]
},
{
"cell_type": "code",
"execution_count": 104,
"metadata": {
"collapsed": true
},
"outputs": [],
"source": [
"from dionysus import fill_rips"
]
},
{
"cell_type": "code",
"execution_count": 237,
"metadata": {
"collapsed": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"\n",
"[('add', 'method'), ('append', 'method'), ('index', 'method'), ('sort', 'method')]\n",
"Filtration with 43 simplices\n"
]
}
],
"source": [
"dim_of_allowed_complex = 2\n",
"max_radius = 0.22\n",
"\n",
"f = fill_rips(X, dim_of_allowed_complex, max_radius)\n",
"info(f)"
]
},
{
"cell_type": "code",
"execution_count": 238,
"metadata": {
"collapsed": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"No. 0 : <0> 0\n",
"No. 1 : <1> 0\n",
"No. 2 : <2> 0\n",
"No. 3 : <3> 0\n",
"No. 4 : <4> 0\n",
"No. 5 : <5> 0\n",
"No. 6 : <6> 0\n",
"No. 7 : <7> 0\n",
"No. 8 : <8> 0\n",
"No. 9 : <9> 0\n",
"No. 10 : <10> 0\n",
"No. 11 : <11> 0\n",
"No. 12 : <12> 0\n",
"No. 13 : <13> 0\n",
"No. 14 : <14> 0\n",
"No. 15 : <15> 0\n",
"No. 16 : <16> 0\n",
"No. 17 : <17> 0\n",
"No. 18 : <18> 0\n",
"No. 19 : <19> 0\n",
"No. 20 : <7,15> 0.0911795\n",
"No. 21 : <2,10> 0.0936293\n",
"No. 22 : <13,14> 0.0937617\n",
"No. 23 : <3,18> 0.10823\n",
"No. 24 : <6,19> 0.121911\n",
"No. 25 : <7,9> 0.146531\n",
"No. 26 : <1,14> 0.148711\n",
"No. 27 : <7,14> 0.16263\n",
"No. 28 : <14,15> 0.173995\n",
"No. 29 : <7,14,15> 0.173995\n",
"No. 30 : <13,15> 0.187649\n",
"No. 31 : <13,14,15> 0.187649\n",
"No. 32 : <8,16> 0.188886\n",
"No. 33 : <12,13> 0.192908\n",
"No. 34 : <3,11> 0.195414\n",
"No. 35 : <3,19> 0.20217\n",
"No. 36 : <1,5> 0.211024\n",
"No. 37 : <18,19> 0.213223\n",
"No. 38 : <3,18,19> 0.213223\n",
"No. 39 : <2,12> 0.216758\n",
"No. 40 : <7,13> 0.218737\n",
"No. 41 : <7,13,14> 0.218737\n",
"No. 42 : <7,13,15> 0.218737\n",
"---\n",
"\n",
"[('boundary', 'method'), ('data', 'non-method'), ('dimension', 'method'), ('join', 'method')]\n",
"<7,13,15> 0.218737\n"
]
}
],
"source": [
"for i, s in enumerate(f):\n",
" print(\"No. {0} : {1}\".format(i, s))# simplex, radius at which the simplex appears in the complex\n",
"print('---')\n",
"info(s)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Show the complex ( faces <= dim.2 )"
]
},
{
"cell_type": "code",
"execution_count": 239,
"metadata": {
"collapsed": false
},
"outputs": [],
"source": [
"dim2simpleciesList = [[] for i in range(dim_feature_space + 1)]\n",
"for s in f:\n",
" dim2simpleciesList[s.dimension()].append(s)"
]
},
{
"cell_type": "code",
"execution_count": 240,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"image/png": 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E4S4xr/ja5wChzBXMvXI6Z/51Gs2++Qh27fJ95eecAyNG+KGKzZsHW7RIJVO4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\nwl1EJA4p3EVE4pDCXUQkDincRUTi0P8DDrA8J0/aWAEAAAAASUVORK5CYII=\n",
"text/plain": [
""
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"plt.xlim([-0.05, 1.05])\n",
"plt.ylim([-0.05, 1.05])\n",
"\n",
"ax = plt.subplot()\n",
"for t in dim2simpleciesList[2]:\n",
" tri = plt.Polygon([ X[v] for v in t], fc='g');\n",
" ax.add_patch(tri)\n",
"\n",
"for e in dim2simpleciesList[1]:\n",
" pt = np.array([X[v] for v in e]).T\n",
" plt.plot(pt[0], pt[1], c='r')\n",
"\n",
"plt.scatter(X[:,0], X[:,1], c='b');"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## - Persistent Homology"
]
},
{
"cell_type": "code",
"execution_count": 251,
"metadata": {
"collapsed": true
},
"outputs": [],
"source": [
"from dionysus import homology_persistence"
]
},
{
"cell_type": "code",
"execution_count": 252,
"metadata": {
"collapsed": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"\n",
"[('pair', 'method'), ('unpaired', 'non-method')]\n",
"Reduced matrix with 43 columns\n"
]
}
],
"source": [
"hp = homology_persistence(f)\n",
"info(hp)"
]
},
{
"cell_type": "code",
"execution_count": 253,
"metadata": {
"collapsed": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"No. 0 : \n",
"No. 1 : \n",
"No. 2 : \n",
"No. 3 : \n",
"No. 4 : \n",
"No. 5 : \n",
"No. 6 : \n",
"No. 7 : \n",
"No. 8 : \n",
"No. 9 : \n",
"No. 10 : \n",
"No. 11 : \n",
"No. 12 : \n",
"No. 13 : \n",
"No. 14 : \n",
"No. 15 : \n",
"No. 16 : \n",
"No. 17 : \n",
"No. 18 : \n",
"No. 19 : \n",
"No. 20 : 1*7 + 1*15\n",
"No. 21 : 1*2 + 1*10\n",
"No. 22 : 1*13 + 1*14\n",
"No. 23 : 1*3 + 1*18\n",
"No. 24 : 1*6 + 1*19\n",
"No. 25 : 1*7 + 1*9\n",
"No. 26 : 1*1 + 1*13\n",
"No. 27 : 1*1 + 1*7\n",
"No. 28 : \n",
"No. 29 : 1*20 + 1*27 + 1*28\n",
"No. 30 : \n",
"No. 31 : 1*22 + 1*28 + 1*30\n",
"No. 32 : 1*8 + 1*16\n",
"No. 33 : 1*1 + 1*12\n",
"No. 34 : 1*3 + 1*11\n",
"No. 35 : 1*3 + 1*6\n",
"No. 36 : 1*1 + 1*5\n",
"No. 37 : \n",
"No. 38 : 1*23 + 1*35 + 1*37\n",
"No. 39 : 1*1 + 1*2\n",
"No. 40 : \n",
"No. 41 : 1*22 + 1*27 + 1*40\n",
"No. 42 : \n",
"---\n",
"\n",
"[]\n",
"\n"
]
}
],
"source": [
"# Result\n",
"for i, c in enumerate(hp):\n",
" print(\"No. {0} : {1}\".format(i, c))\n",
"print('---')\n",
"info(c)"
]
},
{
"cell_type": "markdown",
"metadata": {
"collapsed": true
},
"source": [
"### Persistent diagram"
]
},
{
"cell_type": "code",
"execution_count": 260,
"metadata": {
"collapsed": true
},
"outputs": [],
"source": [
"from dionysus import init_diagrams"
]
},
{
"cell_type": "code",
"execution_count": 268,
"metadata": {
"collapsed": false,
"scrolled": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"dim = 0 : Diagram with 20 points\n",
" (0,inf)\n",
" (0,inf)\n",
" (0,0.216758)\n",
" (0,inf)\n",
" (0,inf)\n",
" (0,0.211024)\n",
" (0,0.20217)\n",
" (0,0.16263)\n",
" (0,inf)\n",
" (0,0.146531)\n",
" (0,0.0936293)\n",
" (0,0.195414)\n",
" (0,0.192908)\n",
" (0,0.148711)\n",
" (0,0.0937617)\n",
" (0,0.0911795)\n",
" (0,0.188886)\n",
" (0,inf)\n",
" (0,0.10823)\n",
" (0,0.121911)\n",
"dim = 1 : Diagram with 0 points\n",
"dim = 2 : Diagram with 1 points\n",
" (0.218737,inf)\n",
"---\n",
"\n",
"[('append', 'method')]\n",
"Diagram with 1 points\n",
"---\n",
"\n",
"[('birth', 'non-method'), ('data', 'non-method'), ('death', 'non-method')]\n",
"(0.218737,inf)\n"
]
}
],
"source": [
"diag = init_diagrams(hp, f)\n",
"\n",
"for dim, dg in enumerate(diag):\n",
" print(\"dim = {0} : {1}\".format(dim, dg))\n",
" for d in dg:\n",
" print(\" \",d)\n",
"print('---')\n",
"info(dg)\n",
"print('---')\n",
"info(d)"
]
},
{
"cell_type": "code",
"execution_count": 264,
"metadata": {
"collapsed": false,
"scrolled": true
},
"outputs": [
{
"data": {
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"text/plain": [
""
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"dim = 0\n",
"dionysus.plot.plot_diagram(diag[dim], show = True)"
]
},
{
"cell_type": "code",
"execution_count": 264,
"metadata": {
"collapsed": false,
"scrolled": true
},
"outputs": [
{
"data": {
"image/png": 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"text/plain": [
""
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"dim = 0\n",
"dionysus.plot.plot_bars(diag[dim], show = True)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## - Draw the filtration process"
]
},
{
"cell_type": "code",
"execution_count": 7,
"metadata": {
"collapsed": true
},
"outputs": [],
"source": [
"from collections import defaultdict\n",
"def get_dim2simpleciesList(filt, radius, dim_feature_space):\n",
" dim2simpleciesList_dic = defaultdict(lambda :[])# \n",
" for s in f:\n",
" if s.data <= radius:\n",
" dim2simpleciesList_dic[s.dimension()].append(s)\n",
" size = max(dim_feature_space, max(dim2simpleciesList_dic.keys()))\n",
" dim2simpleciesList = [[] for i in range(size + 1)]\n",
" for dim, simpleciesLict in dim2simpleciesList_dic.items():\n",
" dim2simpleciesList[dim] = simpleciesLict\n",
" return dim2simpleciesList"
]
},
{
"cell_type": "code",
"execution_count": 26,
"metadata": {
"collapsed": false
},
"outputs": [],
"source": [
"def plot_abstract_complex(X, dim2simpleciesList, ax=None):\n",
" #ax.set_xlim([-0.05, 1.05])\n",
" #ax.set_ylim([-0.05, 1.05])\n",
"\n",
" if ax is None:\n",
" ax = plt.subplot()\n",
" for t in dim2simpleciesList[2]:\n",
" tri = plt.Polygon([ X[v] for v in t], fc='g');\n",
" ax.add_patch(tri)\n",
"\n",
" for e in dim2simpleciesList[1]:\n",
" pt = np.array([X[v] for v in e]).T\n",
" ax.plot(pt[0], pt[1], c='r')\n",
"\n",
" ax.scatter(X[:,0], X[:,1], c='b')"
]
},
{
"cell_type": "code",
"execution_count": 9,
"metadata": {
"collapsed": true
},
"outputs": [],
"source": [
"def get_transition_raidus_in_filtration(filt):\n",
" transition_radius = list(set([s.data for s in f]))\n",
" transition_radius.sort()\n",
" return transition_radius"
]
},
{
"cell_type": "code",
"execution_count": 10,
"metadata": {
"collapsed": true
},
"outputs": [],
"source": [
"def get_dim2BDarray(filt):\n",
" hp = homology_persistence(filt)\n",
" diag = init_diagrams(hp, filt)\n",
" return [np.array([[d.birth, d.death] for d in dg]) for dg in diag], hp, diag"
]
},
{
"cell_type": "code",
"execution_count": 11,
"metadata": {
"collapsed": true
},
"outputs": [],
"source": [
"def BDarray2homRank(BDarray, radius):\n",
" if len(BDarray) == 0:\n",
" return 0\n",
" else:\n",
" return np.sum(np.sum(BDarray <= radius, axis=1) == 1)\n",
"\n",
"def get_homology_rank(dim2BDarray, radius):\n",
" return [(dim, BDarray2homRank(BDarray, radius)) for dim, BDarray in enumerate(dim2BDarray)]"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### generate data"
]
},
{
"cell_type": "code",
"execution_count": 215,
"metadata": {
"collapsed": true
},
"outputs": [],
"source": [
"num_data_points = 20\n",
"dim_feature_space = 2\n",
"\n",
"np.random.seed(10)\n",
"X = np.random.rand(num_data_points, dim_feature_space)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Get the VR complex"
]
},
{
"cell_type": "code",
"execution_count": 437,
"metadata": {
"collapsed": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"Filtration with 249 simplices\n"
]
}
],
"source": [
"dim_of_allowed_complex = 10\n",
"max_radius = 0.4\n",
"\n",
"f = fill_rips(X, dim_of_allowed_complex, max_radius)\n",
"print(f)\n",
"\n",
"dim2BDarray, hp, diag = get_dim2BDarray(f)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Compute the Betti numbers"
]
},
{
"cell_type": "code",
"execution_count": 438,
"metadata": {
"collapsed": false
},
"outputs": [],
"source": [
"#radius_list = np.linspace(0.0, max_radius, 200)\n",
"tr = get_transition_raidus_in_filtration(f)\n",
"radius_list = (np.array([0.0] + tr) + np.array(tr + [max_radius])) / 2.\n",
"\n",
"homology_rank_seq = np.array([np.array(get_homology_rank(dim2BDarray, r))[:,1] for r in radius_list])"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Plot"
]
},
{
"cell_type": "code",
"execution_count": 442,
"metadata": {
"collapsed": false,
"scrolled": true
},
"outputs": [
{
"data": {
"image/png": 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3i+QzFXQpCC1bwpe/7Ge6iOQrFXQpGBoYlXyngi4Fo7wc/vY3qKwMnUQkGiroUjBOPtnf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\nZ66qGgFNAZfsuYF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"text/plain": [
""
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"print(len(radius_list))\n",
"ax = plt.subplot()\n",
"ax.get_yaxis().set_major_locator(MaxNLocator(integer=True))\n",
"ax.plot(radius_list, homology_rank_seq[:,0], c='b', label='0')\n",
"ax.plot(radius_list, homology_rank_seq[:,1], c='r', label='1')\n",
"ax.plot(radius_list, homology_rank_seq[:,2], c='g', label='2')\n",
"ax.legend(loc='upper left');"
]
},
{
"cell_type": "markdown",
"metadata": {
"collapsed": true
},
"source": [
"### Create the animation\n",
"* Warning : It takes a lot of time if you have many simplicies at last\n",
"* You can reduce the radius list to reduce the plot output\n",
" + Ex. : reduce by 1/10\n",
" - radius_list = radius_list[::10]"
]
},
{
"cell_type": "code",
"execution_count": 1,
"metadata": {
"collapsed": true
},
"outputs": [],
"source": [
"import os\n",
"from matplotlib.ticker import MaxNLocator"
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {
"collapsed": true
},
"outputs": [],
"source": [
"save_dir = ...\n",
"os.makedirs(save_dir, exist_ok=True)"
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {
"collapsed": true
},
"outputs": [],
"source": [
"%%time\n",
"fig = plt.figure(figsize=(14,5))\n",
"for i, r in enumerate(radius_list):\n",
" if i % 5 != 0:\n",
" continue\n",
" print(\"{0}/{1}\".format(i, len(radius_list)), '\\r', end='')\n",
" fig.clf()\n",
" ax1 = fig.add_subplot(1,2,1)\n",
"\n",
" ax1.get_yaxis().set_major_locator(MaxNLocator(integer=True))\n",
" ax1.set_ylim([0.,30.])\n",
" ax1.plot(radius_list, homology_rank_seq[:,0], c='b', label='0')\n",
" ax1.plot(radius_list, homology_rank_seq[:,1], c='r', label='1')\n",
" ax1.plot(radius_list, homology_rank_seq[:,2], c='g', label='2')\n",
" ax1.legend(loc='upper left')\n",
" ax1.axvline(x=r, c='y')\n",
"\n",
" ax2 = fig.add_subplot(1,2,2)\n",
" plot_abstract_complex(X, get_dim2simpleciesList(f, r, dim_feature_space), ax2)\n",
"\n",
" fig.savefig(os.path.join(save_dir, '{0:04d}.png'.format(i)), dpi=100)"
]
},
{
"cell_type": "code",
"execution_count": 30,
"metadata": {
"collapsed": true
},
"outputs": [],
"source": [
"import subprocess, os"
]
},
{
"cell_type": "code",
"execution_count": 31,
"metadata": {
"collapsed": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"CPU times: user 4 ms, sys: 44 ms, total: 48 ms\n",
"Wall time: 1.98 s\n"
]
}
],
"source": [
"%%time\n",
"output_dir = ...\n",
"gif_name = \"anim.gif\"\n",
"time_interval = 100\n",
"\n",
"target_frames = os.path.join(save_dir, \"*.png\")\n",
"output_gifname = os.path.join(output_dir, gif_name)\n",
"convert_command = \"convert -layers optimize -loop 0 -delay {2} {0} {1}\".format(target_frames, output_gifname, time_interval)\n",
"result = subprocess.run(convert_command, shell=True)"
]
},
{
"cell_type": "markdown",
"metadata": {
"collapsed": true
},
"source": [
"##### future improvements\n",
"* Relative persistent homology\n",
"* some materials related to more theoretical definitions of PH or PD..."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# 2. Applied examples"
]
},
{
"cell_type": "code",
"execution_count": 1,
"metadata": {
"collapsed": true
},
"outputs": [],
"source": [
"#import dionysus\n",
"from dionysus import Simplex, Filtration\n",
"from dionysus import closure, fill_rips, homology_persistence, init_diagrams\n",
"from dionysus.plot import plot_diagram as dplot\n",
"\n",
"import os\n",
"\n",
"import numpy as np\n",
"import matplotlib.pyplot as plt\n",
"%matplotlib inline\n",
"import matplotlib.cm as cm\n",
"from matplotlib.ticker import MaxNLocator\n",
"\n",
"import seaborn as sns\n",
"sns.set_style('whitegrid')"
]
},
{
"cell_type": "code",
"execution_count": 2,
"metadata": {
"collapsed": true
},
"outputs": [],
"source": [
"def sample_points_from_image(grayscale_image, num_samples=None, ratio=None, threshold=128):\n",
" assert grayscale_image.ndim == 2\n",
" points = np.array(np.where(grayscale_image >= threshold)).T\n",
" np.random.shuffle(points)\n",
" if num_samples is not None:\n",
" return points[:num_samples]\n",
" elif ratio is not None:\n",
" return points[:int(len(points) * ratio)]\n",
"\n",
"def get_1PD(x, num_samples=50, ratio=None, dim_of_allowed_complex=3, max_radius=15.0, threshold=128, dim_list=[1,2]):\n",
" s = sample_points_from_image(x, num_samples=num_samples, ratio=ratio, threshold=threshold)\n",
" f = fill_rips(s.astype(np.float32), dim_of_allowed_complex, max_radius)\n",
" hp = homology_persistence(f)\n",
" diag = init_diagrams(hp, f)\n",
" return { dim : np.array([[d.birth, d.death] for d in diag[dim]]) for dim in dim_list }\n",
"\n",
"def drop_inf(bd_array):\n",
" return bd_array[~(bd_array == np.inf).any(axis=1)]\n",
"\n",
"def plot_PDpoints_for_mnist(points, labels):\n",
" upper = 1.1 * np.max(points[:,1])\n",
" plt.xlim([0.0, upper])\n",
" plt.ylim([0.0, upper])\n",
" plt.scatter(points[:,0], points[:,1], c=labels, cmap=cm.tab10);\n",
" plt.colorbar()\n",
" plt.plot(np.arange(upper), np.arange(upper), c='black');"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### 2-1. Artificial example\n",
"* Warning : It takes much computational cost. Be care of the memory size !"
]
},
{
"cell_type": "code",
"execution_count": 2,
"metadata": {
"collapsed": true
},
"outputs": [],
"source": [
"def generate_rings(num_data_points, seed=0, a=0.1, s=1.0, c=[0., 0.]):\n",
" np.random.seed(seed)\n",
" theta = 4 * np.pi * np.random.rand(num_data_points)\n",
" r = s * np.exp( 2 * a * np.random.rand(num_data_points) - a )\n",
" return (r * np.array([np.cos(theta), np.sin(theta)])).T + np.array(c)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### generate data"
]
},
{
"cell_type": "code",
"execution_count": 3,
"metadata": {
"collapsed": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"200 2\n"
]
}
],
"source": [
"X = np.concatenate([generate_rings(130, seed=5),\n",
" generate_rings(25, seed=2, s=0.2, a=0.05, c=[1.2,0.]),\n",
" generate_rings(45, seed=3, s=0.35, a=0.3, c=[0.95,0.65])])\n",
"\n",
"num_data_points, dim_feature_space = X.shape\n",
"print(num_data_points, dim_feature_space)"
]
},
{
"cell_type": "code",
"execution_count": 4,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/plain": [
""
]
},
"execution_count": 4,
"metadata": {},
"output_type": "execute_result"
},
{
"data": {
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eZG2tkokfvVGzyZN3/USY1NtBmYD3j59emmzhHst1OslgLiOijPNLaudKKu+PXwCpv8na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\ngwGBiIgAMCAQEZHj/wPZzaw29bDUSAAAAABJRU5ErkJggg==\n",
"text/plain": [
""
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"plt.xlim([-1.5, 2.0])\n",
"plt.ylim([-1.5, 1.5])\n",
"plt.scatter(X[:,0], X[:,1])"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Get the VR complex"
]
},
{
"cell_type": "code",
"execution_count": 5,
"metadata": {
"collapsed": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"Filtration with 5281300 simplices\n",
"CPU times: user 16.5 s, sys: 272 ms, total: 16.8 s\n",
"Wall time: 16.7 s\n"
]
}
],
"source": [
"%%time\n",
"dim_of_allowed_complex = 4\n",
"max_radius = 0.6\n",
"\n",
"f = fill_rips(X, dim_of_allowed_complex, max_radius)\n",
"print(f)"
]
},
{
"cell_type": "code",
"execution_count": 12,
"metadata": {
"collapsed": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"CPU times: user 46.7 s, sys: 392 ms, total: 47.1 s\n",
"Wall time: 47 s\n"
]
}
],
"source": [
"%%time\n",
"dim2BDarray, hp, diag = get_dim2BDarray(f)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Compute and plot the Betti number transitions"
]
},
{
"cell_type": "code",
"execution_count": 13,
"metadata": {
"collapsed": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"CPU times: user 4.94 s, sys: 0 ns, total: 4.94 s\n",
"Wall time: 4.94 s\n"
]
}
],
"source": [
"%%time\n",
"radius_list = np.linspace(0.0, max_radius, 60)\n",
"\n",
"homology_rank_seq = np.array([np.array(get_homology_rank(dim2BDarray, r))[:,1] for r in radius_list])"
]
},
{
"cell_type": "code",
"execution_count": 24,
"metadata": {
"collapsed": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"60\n"
]
},
{
"data": {
"image/png": 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"text/plain": [
""
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"print(len(radius_list))\n",
"ax = plt.subplot()\n",
"ax.get_yaxis().set_major_locator(MaxNLocator(integer=True))\n",
"#ax.set_yscale('log')\n",
"ax.set_ylim([0,30])\n",
"ax.plot(radius_list, homology_rank_seq[:,0], c='b', label='0')\n",
"ax.plot(radius_list, homology_rank_seq[:,1], c='r', label='1')\n",
"ax.plot(radius_list, homology_rank_seq[:,2], c='g', label='2')\n",
"#ax.plot(radius_list, homology_rank_seq[:,3], c='y', label='3')\n",
"ax.legend(loc='upper left');"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### 2-2. MNIST example\n",
"* Not good example, however. I'll improve this part in the future and please inform me if you have any idea !"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Note:\n",
"* 0, 6, 9 has single 1-cycle\n",
" - the cycle in 0 may be greater than that of 6 or 9\n",
"* 8 has the two 1-cycles"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"#### Load mnist"
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {
"collapsed": true
},
"outputs": [],
"source": [
"# describe by yourself a MNIST loading line\n",
"x_mnist = ...\n",
"y_mnist = ...\n",
"assert x_mnist.shape[1:] == (28, 28) and len(x_mnist) == len(y_mnist)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"#### check : sampling from digit pixels"
]
},
{
"cell_type": "code",
"execution_count": 16,
"metadata": {
"collapsed": false,
"scrolled": true
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"digit : 5\n",
"# of sampled = 58\n"
]
},
{
"data": {
"image/png": 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"text/plain": [
""
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"# check mnist images and their pixel sampling examples\n",
"i = 0\n",
"print(\"digit : \", y_mnist[i])\n",
"\n",
"plt.figure(figsize=(8,4))\n",
"plt.subplot(121)\n",
"plt.imshow(x_mnist[i], cmap=cm.gray);\n",
"\n",
"s = sample_points_from_image(x_mnist[i], ratio=0.5, threshold=100)\n",
"print(\"# of sampled = \", len(s))\n",
"img = np.zeros(x_mnist[i].shape, np.uint8)\n",
"img[s[:,0], s[:,1]] = 255\n",
"plt.subplot(122)\n",
"plt.imshow(img, cmap=cm.gray);"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"#### Compute the persistent diagram ( birth-death time of cycles )"
]
},
{
"cell_type": "code",
"execution_count": 5,
"metadata": {
"collapsed": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"completed\n",
"CPU times: user 6min, sys: 1.13 s, total: 6min 2s\n",
"Wall time: 6min 1s\n"
]
}
],
"source": [
"%%time\n",
"num_samples = None\n",
"ratio = 0.5\n",
"threshold = 100\n",
"\n",
"np.random.seed(10)\n",
"target = np.arange(100)\n",
"bd_list = []\n",
"for i, n in enumerate(target):\n",
" print(\"{0}/{1}\".format(i, len(target)), '\\r', end='')\n",
" bd_list.append(get_1PD(x_mnist[n], num_samples=num_samples, ratio=ratio, threshold=threshold))\n",
"print('completed')"
]
},
{
"cell_type": "code",
"execution_count": 7,
"metadata": {
"collapsed": false
},
"outputs": [],
"source": [
"from collections import defaultdict\n",
"\n",
"points = []\n",
"labels = []\n",
"cycle_lifetimes = defaultdict(lambda :[])\n",
"for label, bd in zip(y_mnist[target], bd_list):\n",
" if len(bd[1]) != 0:\n",
" cycle_lifetimes[label].append(bd[1][:,1] - bd[1][:,0])\n",
" for pt in drop_inf(bd[1]):\n",
" points.append(pt)\n",
" labels.append(label)\n",
" else:\n",
" cycle_lifetimes[label].append([])\n",
" \n",
"points = np.array(points)\n",
"labels = np.array(labels)\n",
"cycle_lifetimes = { label : np.array(lt) for label, lt in cycle_lifetimes.items() }"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"#### Remove the noise cycles"
]
},
{
"cell_type": "code",
"execution_count": 8,
"metadata": {
"collapsed": false
},
"outputs": [],
"source": [
"delta = 2.\n",
"robust_betti_numbers = defaultdict(lambda :[])\n",
"for label, lt_for_each_digit in cycle_lifetimes.items():\n",
" for lt_list in lt_for_each_digit:\n",
" if len(lt_list) == 0:\n",
" robust_betti_numbers[label].append(0)\n",
" else:\n",
" robust_betti_numbers[label].append(np.sum(lt_list > delta))\n",
"\n",
"robust_betti_numbers = { label : np.array(rbn) for label, rbn in robust_betti_numbers.items()}"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"#### Show the persistent diagrams at the same time\n",
"* Notice that one sample can have several points in the PD below. Here we do not pay attentions to it."
]
},
{
"cell_type": "code",
"execution_count": 9,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"image/png": 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3TuH30/vx9+7AC/3fw9vLh5advHhg3B11buRWlz57S6T+8o9dn1Uq3Fq2bMnx\n48fp378/CQkJODs7lxts4uaWjwshKauAuVtO4ajTMGPAHVVySjJ3b8I/gu26wrOZFPyVjlNQ49t+\nDXHViRMnGDhwIBcvXmTEiBHMfvP/OBsdS48H7sSrhautyxOiQalUuI0cOZIZM2bw9NNPYzQamTVr\nVhWX1TA1c3dkwai7qvSYxal6yxsUKLqc06DDTVEUjGYjWrX2tkewW7ZsYeTIkeTk5DBz5kzeeecd\n1Go1ueZkCTYhbKBS4ebs7Mynn35a1bWIaqAuZ2aeyqFhztpTFIXvTnzHzxd/5kr+FbydvOnn348X\nOr9Q4ZBTFIWFCxfyyiuvoNPp+P777xk1alQ1VS6EuFUN89utAXHs5EXBiXRUN5yZVDeyw6VHw7zX\nannMcj4/+jkmxQRAuiGd0xmnKTAW8J/g/9zycYqLi5k8eTJLlizBx8eHTZs20b179+oqWwhRAbK2\nZD3nfLcP+R20qJ2vX7/TNnbAfWBrNM7Wb4Csr4xmIz9d+Kkk2P6moLD10lYKjJZvybhRRkYGDz/8\nMEuWLKFLly4cPnxYgk2IWkRGbg1A/p12tH08iILoNFQOGpy7NkGla5h/12QYMojLjbO4LT4vntic\nWDp4dij3GGfOnGHgwIGcPXuWIUOGEBYWhouLrAEqRG3SML/hGiBtI3tc72uOyz1NG2ywAbjZueHp\n4Glxm4e9Bz5O1teqA9ixYwfdu3fn7NmzTJ8+nfXr10uwCVELNdxvOdEgOWgd6NW8l8VtPX174u7g\nbnXfxYsX079/f/R6Pd9++y0ffvgh6mpcp1IIUXlyWlI0ONNDpmMwGvg1/leyi7Jx1bnS07cn74S+\nY7G/0Whk6tSpLFy4EG9vb8LDw7n33ntruGohREVIuIkGx05jxwf3fUBiXiIn0k/Q3rM9/q7+Fvtm\nZWUxcuRIfvnlF4KCgoiMjCQgIKBmCxaijnHwqNk1XS2RcBN1jqIoZHz7Lbm/bMOYkYGuRXM8hg7D\n7dFHKnQcXxdffF18rW4/f/48AwcO5NSpUwwYMIDVq1fj5uZ2u+ULIWqAhJuoc1IXLCB92XIwXZ3O\nX3zpEgVHjmIuNOD++ONV8hp79uxh6NChZGRkMHXqVObNmydLzAlRh8jV8EowGs0cvJDOueTc2z5W\nco6BhKwCFKXs+o+iLFNePtkRkSXB9jdFryfrx7VlPsdifTEZpzMwZBdyI4PRQGxOLPri0kuULV++\nvORxTkuQTZOWAAAgAElEQVSXLmX+/PkSbELUMTJyq6A3N0QTfjQBQ7EZgMYudiwY2ZVe7bwrdJxj\nsVl8/MtpjlzOxKiY6dzcnZf6tKFPYPlT0Ru6gqNHMCYlWdxWeOkSSkEBKicnzEYzJxcdQ5eUj4Oi\nkKFSYfCwJ/Dlu1HZq/jkyCfsuLyDxLxEfJx86O3Xm1eDX+XtN99m/vz5eHp6sn79eu6///6afYNC\niCohI7cK+HzHGb4/FFcSbABpeUW88F0UhiLjLR8nW1/EKz8eY9+5NPTFJoqMCn9czuSN9X9ytgpG\ng/WZrnlzVA4OFrdp3NxQ2dsDcGpJNG6JeTgBapUKB8A9s5DTC47w6dFP+favb4nPi8eMmSR9EquO\nraLL/V2YP38+gYGBHDx4UIJNiDpMwq0CVh20vLKFvtjEzMgTt3ycb367xMW0/DLtKbmFhP1+udL1\nNQT2rVvjdM89Frc5h4ai0mgwGc2o43MtLoLskGXg9zP7S7UVpRZxYc4Fzvx2hgf6PcCBAwdo27Zt\ntdQvhKgZEm4VkGMotrrtXEreLR8nKdtgdVtyjvVt4qpms2bi1L07aK+eVVc5OeH60EP4TH8DgMKs\nQuzMlq9h2qtUOGRdv36Wfzaf8++dpzC+EM8HPPl01ae4u1u/kVsIUTfINbcKcHPUkV9ksritbZNb\nX4Kpmbvl02oATd2sbxNX6Zo3x/+bFegPHKDw/HmcgoNx6NixZLu9uz1FahV2FvKtUFEwuJugCDL3\nZ5K4IhHFrNBsTDM6DupIS/eWNfhOhBDVRUZuFfB0dz+L7U46De8O6mhxmyXjQgNo1di5TLuPmz1j\nQuXL9VaoVCqce/bEc8yYUsEGoNGqMbdwtTgD1eDuQEjbUK6su0LC0gRUOhUBUwPwesCL3i1642In\n60QKUR9IuFXAS33b82SIH47/WHi4sYsdy57phoPdrQ+CGznZ8cnIrvRq2xgnOw32WjX3BHjw0dA7\nadtEntpcFQJf7Exuc1f0gFlRMABZHvY0H9+efR/sI21zGk7NnGg3sx3turdjVIdRTO8+3dZlCyGq\niJyWrKAPh3bm/cFBHInLxMvFnjbelftLv6ufOyvHdycl14DRpNCskUOFnwJdl0VHRxMdHU1ubi6N\nGjWiS5cudOrUqcqOr1ar6fjyXRTri8lLyMexiSOpOSn0eagPx44do0+fPqxcsxKTowkvBy+cdE5V\n9tpCCNuTcKsErVZNSCuvKjlWE9eGd43twIED7NixA6Px6u0TycnJXLx4EYPBQHBwcJW+ls5Jh0c7\ndw4dOsRjjz3GlStXeOGFF/j888/R6XQ3P4AQolYxGAwMHDiQiRMnMnToUKv95LSkqFEmk4kjR46U\nBNvfiouL+eOPPzCbzVb2rLw1a9bQu3dvUlJSWLBgAYsXL5ZgE6KO+vLLL2nUqNFN+0m4iRqVnp5O\namqqxW2pqank5d36LRU3YzabmTlzJk8++SQ6nY7NmzczZcqUBnX6V4j65Pz585w7d+6WFliQcBM1\nysnJCTs7O4vb7O3tsb+2wsjt0uv1PPnkk7z33nu0atWKAwcO8MgjFXtqgBCidpk7dy7Tp9/axC8J\nN1GjXFxcaNWqlcVtAQEBVRJuiYmJ9O7dmx9//JH77ruPQ4cOVelkFSFEzdu4cSNdu3bFz8/yLVk3\nkgklosY9+uijGAwGYmNjURQFtVpNy5YtefTRR2/72EeOHGHw4MEkJCQwbtw4Fi9eXGWjQSGE7eze\nvZu4uDh2797NlStXsLOzo2nTpvTs2dNifwm3ekwxGtEfPYbq4gWUu++uNdeaGjVqxLhx4zhz5gyp\nqak0bdqUNm3aYMrJIW/ffuxatsTOr0VJ/+JCM7En0mnk7Ugjb+tT9tevX8+YMWMwGAzMmzeP1157\nrda8ZyHE7VmwYEHJfy9cuJDmzZtbDTaQcKu3sjZuJH3FCopOn8FBrebS+g00+c8UnENDbV0acHWF\nkQ4dOtChQwcUs5mUDz8iZ8sWjCkpqJ2dcQrtQdNZ73JwVwYnfs8jKv84OnsNzTu402fMHTi5Xr9u\npygKH3zwAW+//TbOzs5s3LiRwYMH2/DdCSFsTcKtHtIfPUryR3MxZ2UBoDKbMRw/TuLb/6XV2rVo\nPT1sXGFpaYu+IOO770p+Nufnk7d9B3uM7TlNUEl7caGJS9Hp7Pz2JAMndQGu3vMyfvx4Vq1ahb+/\nP5GRkXTu3LnG34MQouZMnjz5pn1kQkk9lLV+Q0mw/ZMxIYGMVatsUFH5cnfssNiekGd5df6EM5mk\nxeeSnJxMnz59WLVqFaGhoRw6dEiCTQgBSLjVS6aMjHK2pddgJTenmM2Y0svWpABF2rKLSwMYi8zs\n332IkJAQfv/9d5566il27tyJj488xVwIcZWEWz2k821mdZudv38NVmKdoiic++MghyLXk+7rw43r\n96sAJ2O2xX1PJh3kqQmPERsby+zZswkLC8PBytO5hRANk1xzq4c8xo4lb9duihMSSrXb3xGIx6hR\nNqrqupy0VP73+ccknj6JYjajQoVn2+Z0vZiEven68lutvPVEG1WYiq9Gn6Io7Dj+I5sOLcXBwYF1\n69YxbNgwW70NIUQtVulwi4iIYNmyZWi1Wl5++eVbWg6lLlEUhQ1HE9h5MpkraRl0Tz3FC/e1xt3Z\n8uoaNSkn508SEr7HUHgFe3sfmjcfRSO3LiXb7f39aTZvLulffYXhzxiKFYVG99yD96tTUTs6VkkN\nxsxMMr5egeHMGdSOjrj26YPb4EG3NPV+54rFFFzIINjzQRw1LuiNuZzXHeN0R2e6xqWg9fDA5V/3\nEfjav3H7LYU/tp2lMM/E6t2f8Ovxn2jevDkRERHcfffdVfJehBD1T6XCLTMzk0WLFrF+/Xr0ej0L\nFy6sd+H2buQJwg5cwnTtfFlU0nl+PZvKN8+G0NjFdjcFp6Rs4/TptykqTitpS0vbTmCH92nS5OGS\nNufgYJyXLMFcUMDR48fp1KNHldVQnJJC3IQXKTxxoqQtd/t2Ck6coOmb5S+Nk5+Viel8Afc1GYbD\nP66p+Tq15c+Cvfh/sRwHTw9UGg0AQb2bc6X4JO/Oepd9x/fRrVs3Nm3ahK+vb5W9HyFE/VOpa24H\nDhwgNDQUFxcXmjRpwvvvv1/VddnUyaQc1v4RVxJsf4tJyGHRrnO2KYqro8nYuKWlgg2guDiDy7FL\nLT55Wu3oCFW8An7akq9KBRsARiPZ69dTeOFCufsW5OTQ1qFLqWADcNK60truTowaSoIN4MSJE4wb\nN459+/fxxBNPsGfPHgk2IcRNVSrc4uPjMRgMTJgwgdGjR3PgwIGqrsumtsRcIb/IZHFbTEI2CQVF\nLIlL4YekdIqq4REt1hQWXiE39y+L23JyYjAY4mukDsONwXaNOS+PnC1byt3XReeBh31Ti9s87Jvi\nqLn+8Neff/6Z0NBQEhISmDlzJt9//z1OTvJQUSHEzVX6mltWVhaff/45iYmJjB07ll27dpV7vSUq\nKqqyL1XjUpItP3ZFAc64K/T5/S9yuPpePz4Ty3P2Zu6pgceDmc05mM3WPmMNMTGnUKtTLG6tys/f\nvqAAjZVtiSkpxJbzWmq9GU9V2REmgKJWEX0iBrPu6jPYPvnkE7RaLbNnz+bhhx/m6NGjVVB9zatL\nv/uWSP22Vdfrt5VKhZuXlxd33XUXWq0Wf39/nJ2dycjIwMvL+tOpq/oJy9XJv52BbZf2kpZXVKrd\n1NyJlCbOpaatx5lVrMCRMV064Ky19pVfdY4dCyE9Y0+Zdg+Pe7j7rgct7hMVFVWln39K376knzpV\npl3TpAmdpkxB61H+Cij7DqwjIKPsPWnxXun06NaLSZMm8dVXX9G0aVM2btyIVqutU78//1TVn31N\nk/ptqzrrr87Q7Lv7pUrtp39+ZZXVUKnTkr169eL333/HbDaTmZmJXq/H4yZfaHWJt5sDLz/QDnen\n68MxtQo827ihWBg4XS4oIiyxZm6ObtPmdZyd25dqc3JqS9u202rk9QEaT/g3zvffD+rrvz4ad3e8\nX5p402BLL0jnS68fuGyXWKr9gn08n9mv5MH+D/LVV1/RtWtXDh06RPfu3avjLQgh6rlKjdx8fHzo\n378/TzzxBABvv/02anX9uh98bGgA/2rnzQ+H47ickMigHnfwZWEeKTn5FvtnGY01UperayD3dNtA\nfMJqDAXxODg0o0WLp9FoKn8tymg28lX0V/ye9DsFxgLau7fn2aBnaevR1mJ/tZ0dfl8sIveXX9BH\nHUHt6Ij78GFlbhDPvXyJU2++ARcvowKU1q1wfnsS0dpT/KfV/2NAxn14Gz24oksj3LCdc3MuUHil\nkCFDhhAWFoaLi4vF1xdCiJup9DW3UaNGMaoW3BBcnQIaO/PGI4FEReUTHNSM7afiOGQh3OxU0L1R\nzX0RazSOtPR/rsqO9+beN9ly6fpEkFMZpziaepTP+35Oa/fWFvdRqdW4Pfwwbg8/bHF7YVYmp0aN\nwiXzH6uMRB0j77nX6PByS04XXmZ94+0A5P2VR+yiWMx6M9OnT2fOnDn17o8lIUTNkm+QCnjBrzEB\nDmVv4u7j6cb9nq42qOj2/ZH8B7vidpVpj8uN47u/vrOwx605Oeud0sF2jUtGFmN3OWKnvvo5ZuzM\n4NL8SyhFChPnTuTDDz+UYBNC3Db5FqmADs6OLAsKYFgTdzo429PV1ZGX/Lz5Kiigzj4U82DSQQpN\nhRa3ncuu/D19Reet3+/WNNnA9G7TMYebSfwuEQdXB+aumsui1xdV+vWEEOKfZG3JCgpydWJRpwBb\nl1FlXO2sjzidrazKfytU5SzzlafWsOw/yzjxywmCgoKIjIwkICCg0q8lhBA3knArR1aRkVVJ6Zwz\nANn5BDe6+mV/+MyvnLn8CxqNMw8EP4d3oyaVOn5SXhIR5yMwKSYebvUwrRtZvr51I0UxkZwcSb7+\nIk6OAfj4DEKtrtz/yuHthvP9ye+Jzyt9A7gKFfe1uK/cfQ9l57E7PRcXjZqnmzfG7R+3QjQe8QS5\nMbPQmUvf03bBaOQ/h6M4Fx/PgAEDWL16NW5ubpWqXQghrJFws2JDcgZzzieRUFgMaNh47BwDvdwI\njp1DE+0+mmqLAfj1t/XYeUxkUI//r0LH/+avb/j6z6/JLMwEIOxEGMPbD+fVbq+Wu19BQTwxf/2H\nnJzrNzTHJ4TRseMnODu1rNibBJx0Trx+z+vM/2M+l3MvA+Cic+GRVo/w1B1PWdzHpCi8fDKWn1Kz\nMFwLr+UJabzTxpfHfK7eCtBqxBMc/W0/5u07sS++OpN0f1EhryUlkW0w8OqrrzJ37lw0muq/N1AI\n0fBIuFmQVWRk9vlEEguvT+8vMCvkpqykuf0u1P+4vOZun0lW5iJSswfe8gjuZPpJlhxfQl7x9ZVQ\n8orzWHliJV28u9CvZT+r+545O7tUsAHk5Bzn3NnZdOmy9BbfYWl9/PsQ6htKxPkIcotyud/vftq4\nt7Ha/4vYFNYnZ5ZqSygs5v3ziTzg5YbLtRHcXZ98Ss6FC1z4YiHhMX/x4c7dKIrCsmXLeO65qpvt\nKYQQN5IJJRasSkovFWx/68yxUsH2N3f7LHZE3XqwRJyPKBVsfzMqRnbE7rC6X3FxFllZhy1uy8w6\nTFFRmsVtt8JB68ATHZ7guTufKzfYAPZk5Fpsjy8sZtUNN7M7t2zJSjS8t3Ubrq6ubN++XYJNCFHt\nZORmQb7J8mLIdlieVQhgMulv+fjWZicCGIyGcl/D2uuYTHqMxnzs7Brfch2VZShnsei8f3x2ubm5\njB49ms2bNxMYGMjmzZtp06b84BRCiKogIzcL+nq54WhhiBaH5WtaRUYdbfysn0q8URfvLla3BXoG\nWt1mb98UF5cOFre5uATi6Oh3yzXcjjucHSy2O6lVPOh1dfblpUuX6NmzJ5s3b+ahhx7iwIEDEmxC\niBoj4WZBt0bODGriXqY92m44KYayj2tJLu5Bj8A+t3z8ga0Hcq/vvWXaOzfuzJiOY6zup1KpadHi\nGdRq5xvanfBrMQaVqnL/O81mM8kLwzg/9DnOD3ia2EnvUJiYarX/RP8mtHIo+xiEx5q409nNmf37\n9xMSEkJMTAyTJ0/mp59+wt297OcphBDVRU5LWrEg0J87nB3Yk5FLSnYO9zT15nFHHWePlX7Om8kM\nRcW3fkoSQKPW8GnfT1kavZQjKUcwm810atyJ5zs/j5Ou/DUitVpXVKrSMwzVag1abeVXSIn79wz0\neyPg2vMOis5Hcfn4H/ivWo6Df7My/Z3UGuxuWEVEBbhpNYSFhTF+/HhMJhNffPEFL774YqXrEkKI\nypJws0KtUvGivw8v+vtcfexEBz+WbhpNa9fSIxqNGnwdY/j9r4306DTE6vHyivTMi/mFo/kKapVC\niKuW1+58AXvt1WWozEUmcvfEkxofj0qtwr5NI1xCm6PSXD89qigKsbFfYTLllDq2yZTL5dhleHv3\nt7hSiqrITNbWSxQn5KHSqnFo74FzSFNUahV5v0ejP7AVKH0/min1IsnvLaDlsrlljvdpbDKn9aWv\nG5rNZha+O4vMVctxd3dn7dq19Ot366dqhRCiKkm4VYCz+YrFdkddITEXrIdbgbGA4b//wjFTQEnb\nwQz44/efWNtzEBqTirQVMRRdvB5ahpMZFF3OwXP0HSWBVVh4haycGCwt9JWd8ycGQ3yZ624mfTHu\newvJy4i7fuwT6RTF5eA5ogNZP0aClUksRRdOW2yPzi09UlUKCsj+8G0K9+2kcUBr9m35Hx06WL42\nKIQQlVFQUMD06dNJT0+nsLCQiRMn0qeP9ctBcs2tApoZLU+kAPDUW19u6tO/tpcKtr8dKGrJd2d3\nkbsvoVSw/a0gJp2Cv65PrVepdBQqlm96LkaDSlX2OljunnjsMso++Vp/PJXCC1mgK+cR4lZWPdH9\nY3RoSrlCxpRnKdy3E13Xe3ht408SbEKIKrdr1y6CgoJYuXIlCxYs4KOPPiq3v4RbBbhm3mGxXV3o\nSpe0x6zudzSv2PIGlZr9GTkUxZe95w0ABQrPXb9ZOl1x4wyWg+M0gSQrZR8UWpxo5dhGhYJTGTSe\n8DQ4NLLYxeHOrhbbe7hffbxP8ckYMiaOwXjuNI4DhxG4YAnP3SEzIoUQVe/RRx/l+eefByApKQkf\nH59y+0u4VYBP0mM4J98N5usjF3WRE14XBuNq52t1P5267Mjpb3ZqSl1Xu5FKo/5HXzWb1M8QS+mH\ngsbhx0b1WBwsPCrmZsd2aOWL+4hnwO6fMzBV2LUJpvkHr1nc7z8BPrQ7tIeMV8ZjzsrA9aVp+E97\nh1fb+tHYrpyRoBBC3KZRo0bx2muvMWPGjHL7yTW3CrBv7k7z45PJbRJFgccpVGY7GsXfh31BMxq9\n3Mrqfg95ubMz3ohZVfrj1lHI4838sdd6UhBtYXURnRrHrt4lPza209LcoyOz0j/kAbbSmBTSacx2\nHqanexN87MsGi31bdwynMsu0q5w0OAVf/cun2Vsv4vZIH9KXrcJcWIhzSDe8nhuGWlv2FKiiKHz4\n3nvse/ddHF1cGPz5Uu7s+wCjm3nSysn6aVshhKgKa9as4eTJk0ybNo2IiAirjxuTcKuAxmM7cuXj\nKNxSuuGW0q2k3aFzY+x9rU/Ff7ptH37LDOcnfQuKsb+6D3pGuF3hIf+hKH4KRZey0R9NAeO1UZ69\nBtf7mmPvV3rF/P+28WViYTH/y79+GjTQ2YH/tik7ZR/ApWdzrkTH4hhvgmuLh6gcNbj18UfX+Pp1\nQue7A3H+4v1y339BQQHjxo3jxx9/pFWrVkRGRtKpU6dy9xFCiKoQExODl5cXzZo144477sBkMpGR\nkYGXl5fF/hJuFaC209J0+j3kbo+l4FQGaq0at4db4tC67LWuUvup1SzuPoyd8VFEJF1Ao4InWrSj\ne9OeAKhUKjyHtcf5riYUnExHpb46YrNr5lLmWIEujvwU3J6ViWnEGYpo7mDHWN/GOGosn2FWqVXk\nhNjRol9rCs9modKocAr2Qedd/v10N0pMTGTIkCEcPnyY++67jw0bNtC4cfUv9SWEEAB//PEHCQkJ\nvPXWW6SlpaHX6/HwsP7dK+FWQWq1mkYPBdDooYAK79u3RTB9WwRb3W7f2h371jdfycNRo+Z5vwo8\nQ06lwrG9J47tPW99n384cuQIgwcPJiEhgWeffZYvv/wSe3v7Sh1LCCEqY9SoUbz11luMHj0ag8HA\nO++8g9rCPIO/SbiJcq1fv54xY8ZgMBiYN28er732mtVz3EIIUV0cHByYP3/+LfeX2ZKVUGQ2cygr\njzP51lfwt8SsmIlJi+Fk+kkURUFRFHJzT5CdfRxFsb7SflVQjEb0x45RePYs6QXFfB0dx++JWdb7\nKwpz5sxh+PDhqNVqNm7cyLRp0yTYhBB1gozcKmh5fCrfJqRxRl+IvUrFPY2cebetL51cy7+G9cvF\nX1j+13JOpp9EjZp+3s151N2IYrgEmHB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IvtSnSfvFjCc7T3/Ju3veR2fUsXjxYiZPnoxC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\nIvrW9vrQ8/4AAAAASUVORK5CYII=\n",
"text/plain": [
""
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"plot_PDpoints_for_mnist(points, labels)"
]
},
{
"cell_type": "code",
"execution_count": 10,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"image/png": 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St2/fhre3N7KysjBo0CDs3bsX5ubmQseiFoLTkkSkcqdPn4aLiwuysrIQHByMQ4cOsdio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\naZ3/Bzsi4xPhYlPaAAAAAElFTkSuQmCC\n",
"text/plain": [
""
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"delta = 2.0\n",
"robust = points[:,1] - points[:,0] > delta\n",
"\n",
"plot_PDpoints_for_mnist(points[robust], labels[robust])"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"#### Show the betti number distributions"
]
},
{
"cell_type": "code",
"execution_count": 11,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"image/png": 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bpSHWU5TV1EagzKNXy2Z22f3P//wP9u/fj9o5GmoyVOxTFgLlDRq2bduGDz74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\nDMbwJSIiMhjDl4iIyGAMXyIiIoP9/xoQk+65+nM6AAAAAElFTkSuQmCC\n",
"text/plain": [
""
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"sns.violinplot(data=[robust_betti_numbers[l] for l in range(10)]);"
]
}
],
"metadata": {
"kernelspec": {
"display_name": "python3free",
"language": "python",
"name": "python3free"
},
"language_info": {
"codemirror_mode": {
"name": "ipython",
"version": 3
},
"file_extension": ".py",
"mimetype": "text/x-python",
"name": "python",
"nbconvert_exporter": "python",
"pygments_lexer": "ipython3",
"version": "3.6.0"
}
},
"nbformat": 4,
"nbformat_minor": 2
}