{ "metadata": { "name": "" }, "nbformat": 3, "nbformat_minor": 0, "worksheets": [ { "cells": [ { "cell_type": "heading", "level": 1, "metadata": {}, "source": [ "Testing a Quasi-Isometric Quantized Embedding" ] }, { "cell_type": "heading", "level": 2, "metadata": {}, "source": [ "Introduction:" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$$\n", "\\newcommand{\\bs}{\\boldsymbol}\n", "\\newcommand{\\cl}{\\mathcal}\n", "\\newcommand{\\bb}{\\mathbb}\n", "$$ \n", "This simple *python notebook* aims at estimating the validity of some of the results explained in the paper \n", " \n", "> \"*A Quantized Johnson Lindenstrauss Lemma: The Finding of Buffon's Needle*\"
\n", "> by [Laurent Jacques](http://perso.uclouvain.be/laurent.jacques/) [(arXiv)](http://arxiv.org/abs/1309.1507)

\n", "> \"In 1733, Georges-Louis Leclerc, Comte de Buffon in France, set the ground of geometric probability theory by defining an enlightening problem: What is the probability that a needle thrown randomly on a ground made of equispaced parallel strips lies on two of them? In this work, we show that the solution to this problem, and its generalization to N dimensions, allows us to discover a quantized form of the Johnson-Lindenstrauss (JL) Lemma, i.e., one that combines a linear dimensionality reduction procedure with a uniform quantization of precision $\\delta>0$. In particular, given a finite set $\\cl S \\in \\bb R^N$ of $S$ points and a distortion level $\\epsilon>0$, as soon as $M > M_0 = O(\\log |S|/\\epsilon^2)$, we can (randomly) construct a mapping from $(\\cl S, \\ell_2)$ to $((\\delta \\bb Z)^M, \\ell_1)$ that approximately preserves the pairwise distances between the points of $\\cl S$. Interestingly, compared to the common JL Lemma, the mapping is quasi-isometric and we observe both an additive and a multiplicative distortions on the embedded distances. These two distortions, however, decay as $O(\\sqrt{\\log |S|/M})$ when $M$ increases. Moreover, for coarse quantization, i.e., for high $\\delta$ compared to the set radius, the distortion is mainly additive, while for small $\\delta$ we tend to a Lipschitz isometric embedding. Finally, we show that there exists \"almost\" a quasi-isometric embedding of $(\\cl S, \\ell_2)$ in $((\\delta \\bb Z)^M, \\ell_2)$. This one involves a non-linear distortion of the $\\ell_2$-distance in $\\cl S$ that vanishes for distant points in this set. Noticeably, the additive distortion in this case decays slower as $O((\\log S/M)^{1/4})$.\" \n", "\n", "\n", "In particular, our objective is to study the behavior of a nonlinear mapping from $\\bb R^N$ to $\\bb R^M$, for two specific dimensions $M,N \\in \\bb N$. This mapping is defined as follows. \n", "\n", "Let a random (sensing) matrix $\\bs \\Phi \\in \\bb R^{M \\times N}$ be such that each component $\\Phi_{ij} \\sim_{\\rm iid} \\cl N(0,1)$, or more shorthly $\\bs \\Phi \\sim \\cl N^{M\\times N}(0,1)$.\n", "\n", "We want to combine the common random projection realized by $\\bs \\Phi$ with the following non-linear operation, i.e., a uniform scalar quantizer of resolution $\\delta > 0$:\n", "
\n", "\n", "$$\n", "\\qquad\\cl Q_\\delta[\\lambda] = \\delta \\lfloor \\lambda /\u00a0\\delta \\rfloor. \n", "$$\n", "
\n", "(remark: we could define instead a *midrise* quantizer by adding $\\delta/2$ to the definition above, and the rest of the development would be unchanged) \n", "\n", "This combination reads\n", "
\n", "\n", "$$\n", "\\qquad\\bs \\psi_\\delta(\\bs x) := \\cl Q_\\delta(\\bs\\Phi \\bs x + \\bs \\xi)\n", "$$\n", "
\n", "with $\\bs \\xi \\sim \\cl U^M([0, \\delta])$ is a *dithering* vector, i.e., a random uniform vector in $\\bb R^M$ such that $u_i \\sim_{\\rm iid} \\cl U([0, \\delta])$, and $\\bs \\psi_\\delta:\\bb R^N \\to (\\delta \\bb Z)^M$ $is our mapping of interest.\n", "\n", "In Proposition 13 (page 16) of the paper above, it shown that\n", "\n", "> **Proposition 13:** Fix$\\epsilon_0>0$,$0< \\epsilon\\leq \\epsilon_0$and$\\delta>0$. There exist two values\n", "$c,c'>0$only depending on$\\epsilon_0$such that, for\n", "$\\bs\\Phi \\sim \\cl N^{M\\times N}(0,1)$and$\\bs \\xi\\sim \\cl U^M([0, \\delta])$, both determining the mapping\n", "$\\bs\\psi_\\delta$above, and for$\\bs u, \\bs v\\in\\bb R^N$,\n", " \n", "$$\\qquad(1 - c\\epsilon) \\|\\bs u - \\bs v\\| - c'\\epsilon\\delta\\ \\leq\\ \\tfrac\n", "{\\sqrt\\pi}{\\sqrt 2 M} \\|\\bs\\psi_\\delta(\\bs u) - \\bs\\psi_\\delta(\\bs v)\\|_1\\ \\leq\\ \n", "(1 + c\\epsilon)\\|\\bs u - \\bs v\\| + c'\\epsilon\\delta.\\tag{1}\n", "$$\n", " \n", "with a probability higher than$1 - 2 e^{-\\epsilon^2M}$.\n", "\n", "This proposition shows that the random quantity$c_\\psi M^{-1} \\|\\bs\\psi_\\delta(\\bs u) - \\bs\\psi_\\delta(\\bs v)\\|_1$with$c_\\psi = \\sqrt{\\pi/2}$*concentrates* around its mean$\\|\\bs u - \\bs v\\|$. However, conversely to linear random projections, this concentration phenomenon displays two kind of deviation: a standard multiplicative one (the$1\\pm c\\epsilon$factor) and an *additive* one in$\\pm c'\\epsilon\\delta$. \n", "\n", "Moreover, forcing (1) to be valid at constant probability, we deduce also that$\\epsilon = O(1/\\sqrt M)$, i.e., showing that both the multiplicative and the additive distortion decay as$O(1/\\sqrt M)$with respect to$M$.\n", "\n", "> *Remark:* From a standard union bound argument (see the paper), this result allows one to show the existence of a non-linear embedding from a set$\\cl S \\subset \\bb R^N$to$(\\delta \\bb Z)^M$, as soon as$M \\geq M_0 = O( \\epsilon^{-2}\\,\\log |\\cl S|)$. Proposition 13 is thus central to reach this property. \n", "\n", "This notebook proposes to test the behavior of the two distortions displayed in (1). \n", "\n", "For that, arbitrarily fixing$\\|\\bs u - \\bs v\\|=1$by normalizing conveniently$\\bs u$and$\\bs v$(e.g., after their random selection in$\\bb R^N$), we study the standard deviation of the random variable\n", " \n", "$$\n", "\\qquad V_\\psi = \\tfrac{\\sqrt\\pi}{\\sqrt 2 M} \\|\\bs\\psi_\\delta(\\bs u) - \\bs\\psi_\\delta(\\bs v)\\|_1.\n", "$$\n", "\n", "> *Remark:* the concentration (1) is invariant under the coordinate change$\\bs u \\to \\lambda \\bs u$,$\\bs v \\to \\lambda \\bs v$and$\\delta \\to |\\lambda| \\delta$for any$\\lambda \\neq 0$, which allows us to set$\\|\\bs u - \\bs v\\|=1$without any loss of generality (providing we study the variability of (1) in$\\delta$). \n", "\n", "Although this random variable displays a non-trivial distribution1, we expect anyway that, if Prop. 13 above is true, \n", " \n", "$$\n", "\\qquad (\\text{Var}\\ V_\\psi)^{1/2}\\quad \\simeq\\quad a\\ \\epsilon\\ +\\ b\\epsilon\\ \\delta,\\quad \\text{for some }a,b > 0,\n", "$$\n", "\n", "and possibly find the constants$a, b>0$by a simple linear fit.\n", "\n", "---\n", " 1: Actually, a mixture of *Buffon* random variable and of a Chi distribution with$N$degrees of freedom. " ] }, { "cell_type": "heading", "level": 2, "metadata": {}, "source": [ "The simulations:\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "> *Remark:* launch ipython notebook with the pylab option, i.e., \n", "> ipython notebook --pylab=inline" ] }, { "cell_type": "code", "collapsed": false, "input": [ "import numpy as np" ], "language": "python", "metadata": {}, "outputs": [], "prompt_number": 1 }, { "cell_type": "markdown", "metadata": {}, "source": [ "Defining the$\\bs \\psi_\\delta:\\bb R^N \\to (\\delta \\bb Z)^M$with two functions:" ] }, { "cell_type": "code", "collapsed": false, "input": [ "def quant(val, d): # the uniform quantizer of resolution 'd'\n", " return d * np.floor(val/d)" ], "language": "python", "metadata": {}, "outputs": [], "prompt_number": 2 }, { "cell_type": "code", "collapsed": false, "input": [ "def psi(vec, mat, dith, d): # the mapping (matrix A, dithering u, quantization d)\n", " return quant(mat.dot(vec) + dith, d)" ], "language": "python", "metadata": {}, "outputs": [], "prompt_number": 3 }, { "cell_type": "heading", "level": 3, "metadata": {}, "source": [ "Entering into the simulations: (time for a coffee/tea break)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "*Experimental setup:* The standard deviation of$V_\\psi$is evaluated for several number of measurements$M$and for several values of$\\delta$. For each of them a certain number of trials are generated. We found that for small$M$a higher number of trials leads to more stable estimation and we set arbitrarily this number to$1024*64/M$(setting this number to 100 also works but less accurately). As for estimating the restricted isometry property in Compressed Sensing, the sensing matrix$\\bs \\Phi$, and the dithering$\\bs \\xi$as well, are regenerated at every trial." ] }, { "cell_type": "code", "collapsed": false, "input": [ "# space dimension\n", "N = 512 \n", "c_psi = np.sqrt(np.pi/2) # used for normalizing psi\n", "\n", "# variable number of dimensions\n", "M_v = np.array([64, 128, 256, 512, 1024])\n", "nb_M = M_v.size\n", "\n", "nb_delta = 8\n", "\n", "# The fixed value for ||u - v|| = 1\n", "dist_l2 = 1 \n", "\n", "# Testing several delta, a logspace arrangement seems logical, here from 10^-3 to 10\n", "delta_v = np.logspace(-3,1,nb_delta) \n", "\n", "V_psi_std = np.zeros([nb_M,nb_delta])\n", "\n", "# Only the standard deviation of V_psi matters here, \n", "# but interested readers could evaluate its mean too to see that it matches ||u - v||. \n", "# In this case, just uncomment this line and the one \n", "# V_psi_avg = np.zeros([nb_M,nb_delta])\n", "\n", "for j in range(nb_M):\n", " M = M_v[j]\n", "\n", " # nb_trials = 100 # uncomment for a constant number of trials\n", " nb_trials = 1024*64/M # better stability to have more trials for small M\n", " V_psi = np.zeros(nb_trials) # temporary store distances in the embedded space\n", " \n", " for k in range(nb_delta):\n", " for i in range(nb_trials):\n", " # local variables for the loop\n", " delta = delta_v[k]\n", " \n", " # Defining the sensing matrix and dithering\n", " Phi = np.random.randn(M,N)\n", " # and dithering, i.e., uniform random vector in [0, \\delta]\n", " xi = np.random.rand(M,1)*delta \n", " \n", " # Generating two points in R^N and normalizing their distance to 1 \n", " tmp_u = np.random.randn(N,1)\n", " tmp_v = np.random.randn(N,1)\n", " tmp_dist_uv = np.linalg.norm(tmp_u - tmp_v)\n", " \n", " u = tmp_u * dist_l2 / tmp_dist_uv;\n", " v = tmp_v * dist_l2 / tmp_dist_uv;\n", " \n", " # projecting the two points according to psi\n", " u_proj = psi(u, Phi, xi, delta);\n", " v_proj = psi(v, Phi, xi, delta);\n", " \n", " # storing one sample of the normalized l1 distance\n", " V_psi[i] = c_psi*np.linalg.norm(u_proj - v_proj,1)/M\n", " \n", " \n", " V_psi_std[j,k] = np.std(V_psi)\n", " # For mean evaluation, you could estimate too: \n", " # V_psi_avg[j,k] = np.mean(V_psi)\n", " \n", " print j, # observing the loop progression (sober progress bar ;-)" ], "language": "python", "metadata": {}, "outputs": [ { "output_type": "stream", "stream": "stdout", "text": [ "0 " ] }, { "output_type": "stream", "stream": "stdout", "text": [ "1 " ] }, { "output_type": "stream", "stream": "stdout", "text": [ "2 " ] }, { "output_type": "stream", "stream": "stdout", "text": [ "3 " ] }, { "output_type": "stream", "stream": "stdout", "text": [ "4\n" ] } ], "prompt_number": 4 }, { "cell_type": "markdown", "metadata": {}, "source": [ "Let's observe now the diferent evolutions of$V_{\\psi}$vs.$\\delta$for the different number of measurements$M\\in \\{64, \\cdots, 1024\\}$selected above: " ] }, { "cell_type": "code", "collapsed": false, "input": [ "fig, ax = plt.subplots()\n", "\n", "for k in range(M_v.size):\n", " ax.plot(delta_v, V_psi_std[k], '.-', label = \"M = %i\" % M_v[k])\n", " \n", "ax.set_title(r'$({\\rm Var}\\ V_\\psi)^{1/2}$vs$\\delta$')\n", "ax.set_xlabel(r'$\\delta$')\n", "ax.set_ylabel(r'$({\\rm Var}\\ V_\\psi)^{1/2}$')\n", "ax.legend(bbox_to_anchor=(1.40, 1))\n", "\n", "fig, ax = plt.subplots()\n", "\n", "for k in range(M_v.size):\n", " ax.plot(delta_v[0:-1], V_psi_std[k][0:-1], '.-', label = \"M = %i\" % M_v[k])\n", " \n", "ax.set_title(r'$({\\rm Var}\\ V_\\psi)^{1/2}$vs$\\delta$(zoom on x-axis)')\n", "ax.set_xlabel(r'$\\delta$')\n", "ax.set_ylabel(r'$({\\rm Var}\\ V_\\psi)^{1/2}$')\n", "ax.legend(bbox_to_anchor=(1.40, 1))\n", " " ], "language": "python", "metadata": {}, "outputs": [ { "metadata": {}, "output_type": "pyout", "prompt_number": 5, "text": [ "" ] }, { "metadata": {}, "output_type": "display_data", "png": 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otti+fTstWrTAw8ODWbNm5dq+atUq2rZti6+vL3369CE2NrbQxwohSi4zExYu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\nxB8l+sOCIVVKCkII0dk0/3TJz8/n6aef9t8uLy9n5syZQfe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sOHv2bKnGCDK3hRCiCNLSYPlyeP996NABZs4ELy9TRyXKisxtoXFzc+P555/n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\nHhRvK3DxPy6i09iJ082n8euaX6O5uxmzp8/G4Vc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"text": [ "" ] } ], "prompt_number": 5 }, { "cell_type": "markdown", "metadata": {}, "source": [ "From this last plot, we conclude that:\n", "\n", "* for$\\delta \\gtrsim 0.2$,$({\\rm Var}\\ V_\\psi)^{1/2}$displays a linear progression in$\\delta$;\n", "* for very small$\\delta$compared to$\\|\\bs u - \\bs v\\|=1$,$({\\rm Var}\\ V_\\psi)^{1/2}$saturates (note that this is **not** explained by the theory); \n", "* for different value of$M$, the linear trend of$({\\rm Var}\\ V_\\psi)^{1/2}$has an offset as$\\delta \\simeq 0$;\n", "* the slope and the offset of the linear progresison clearly depend on$M$. \n", "\n", "Let's go one step further in our analysis.\n", "\n", "From this plot, we perform a linear fit to estimate$(\\text{Var}\\ V_\\psi)^{1/2}$as\n", " \n", "$$\n", "\\qquad(\\text{Var}\\ V_\\psi)^{1/2}\\quad \\simeq\\quad v_\\alpha +\\ v_\\beta\\ \\delta,\\qquad \\text{for some }v_\\alpha, v_\\beta > 0,\n", "$$\n", "\n", "Actually, we have to estimate one couple of$(v_\\alpha,v_\\beta)$for each value$M$, since, from what we explained above, we should have \n", " \n", "$$\\qquad v_\\alpha\\ =\\ a\\ \\epsilon$$ \n", "\n", "and \n", " \n", "$$\\qquad v_\\beta\\ =\\ b\\ \\epsilon$$ \n", "\n", "for some constants$a$and$b$and$\\epsilon = O(1/\\sqrt{M})$depending on$M$(at fixed probability for inequality (1) above).\n", "\n", "Let us see if this is what we get. We first do the linear fit:" ] }, { "cell_type": "code", "collapsed": false, "input": [ "# initialization\n", "v_alpha = np.zeros(M_v.size)\n", "v_beta = np.zeros(M_v.size)\n", "\n", "# For each M, use np.polyfit at degree 1\n", "for k in range(M_v.size):\n", " v_beta[k], v_alpha[k] = np.polyfit(delta_v, V_psi_std[k], 1)\n" ], "language": "python", "metadata": {}, "outputs": [], "prompt_number": 6 }, { "cell_type": "markdown", "metadata": {}, "source": [ "If$v_\\beta = b \\epsilon$, where we recall that$v_\\beta$is the *slope* of each curves in the plots above, we should see a good match between$v_\\beta(M)$and \n", " \n", "$$\n", "\\qquad v^{\\rm th}_\\beta(M)\\ :=\\ \\sqrt{\\tfrac{M_0}{M}}\\ v_\\beta(M_0),\\qquad \\text{with v^{\\rm th}_\\beta(M_0)=v_\\beta(M_0)}\n", "$$ \n", "\n", "for one fixed value$M_0$. \n", "\n", "Is it what we have?" ] }, { "cell_type": "code", "collapsed": false, "input": [ "plt.semilogx(M_v, v_beta, 'o-', label=r'$v_\\beta(M)$')\n", "plt.semilogx(M_v, v_beta[0]*np.sqrt(M_v[0])/np.sqrt(M_v), 'o-', label=r'$v^{\\rm th}_\\beta(M)$') # M_0 is the first M of M_v\n", "plt.xlabel(r'$M$')\n", "plt.ylabel(r'$\\beta$')\n", "plt.axis('tight')\n", "plt.ylim(0, v_beta[0])\n", "plt.legend();" ], "language": "python", "metadata": {}, "outputs": [ { "metadata": {}, "output_type": "display_data", "png": 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XcOWKuasU4v5gsmBJSEjA0dFR/97Z2Zldu3YVWSY+Ph5nZ2f9+4YNG6LVaoss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\nB2zNXd59T4LFAAkWISquvXt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"text": [ "" ] } ], "prompt_number": 7 }, { "cell_type": "markdown", "metadata": {}, "source": [ "The match sounds good enough!\n", "\n", "Finally, if we really have$v_\\alpha = a \\epsilon$and$v_\\beta = b \\epsilon$,$v_\\beta$should be a linear function of$v_\\alpha$, i.e.,$v_\\alpha/v_\\beta \\simeq a/b$.\n", "\n", "Let's test this fact." ] }, { "cell_type": "code", "collapsed": false, "input": [ "fig, ax = plt.subplots()\n", "ax.plot(v_alpha, v_beta, '.-', label = r'$v_\\beta$vs.$v_\\alpha$')\n", "ax.plot(np.linspace(0,v_alpha.max(),10), np.linspace(0,v_alpha.max(),10)*v_beta.max()/v_alpha.max(), label = 'linear trend')\n", "ax.set_xlim(0, v_alpha.max())\n", "ax.set_ylim(0, v_beta.max())\n", "ax.set_xlabel(r'$v_\\alpha$')\n", "ax.set_ylabel(r'$v_\\beta\$')\n", "ax.legend(bbox_to_anchor=(.5, 1))\n", "\n", "ratio, offset = np.polyfit(v_alpha,v_beta,1)\n", "print \"We approximatley have v_beta = %f v_alpha + %f\" % (ratio, offset)" ], "language": "python", "metadata": {}, "outputs": [ { "output_type": "stream", "stream": "stdout", "text": [ "We approximatley have v_beta = 0.357101 v_alpha + 0.000081\n" ] }, { "metadata": {}, "output_type": "display_data", "png": 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j4OrqitWrV193/MHBwTh8+LDx/aFDh657H9R6mXoyCgBqaoB//hN4/XXgrbeAceMUDlQB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\nVhdE5MyYNCzE6oKIiEnDIqw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"text": [ "" ] } ], "prompt_number": 8 }, { "cell_type": "markdown", "metadata": {}, "source": [ "---\n", "

Laurent Jacques, November 2013

" ] } ], "metadata": {} } ] }