arXiv subjects

Mihailo Stojnic

Publications and source records attributed to Mihailo Stojnic.

80 records · Page 5Linked to original sources

arXiv2013

Optimality of $\ell_2/\ell_1$-optimization block-length dependent thresholds

The recent work of \cite{CRT,DonohoPol} rigorously proved (in a large dimensional and statistical context) that if the number of equations (measurements in the compressed sensing terminology) in the system is proportional to the length of the unknown vector then there is a sparsity (number of non-zero elements of the unknown vector) also proportional to the length of the unknown vector such that $\ell_1$-optimization algorithm succeeds in solving the system. In more recent papers \cite{StojnicCSetamBlock09,StojnicICASSP09block,StojnicJSTSP09} we considered under-determined systems with the so-called \textbf{block}-sparse solutions. In a large dimensional and statistical context in \cite{StojnicCSetamBlock09} we determined lower bounds on the values of allowable sparsity for any given number (proportional to the length of the unknown vector) of equations such that an $\ell_2/\ell_1$-optimization algorithm succeeds in solving the system. These lower bounds happened to be in a solid numerical agreement with what one can observe through numerical experiments. Here we derive the corresponding upper bounds. Moreover, the upper bounds that we obtain in this paper match the lower bounds from \cite{StojnicCSetamBlock09} and ultimately make them optimal.

cs.IT↗

arXiv2013

A performance analysis framework for SOCP algorithms in noisy compressed sensing

Solving under-determined systems of linear equations with sparse solutions attracted enormous amount of attention in recent years, above all, due to work of \cite{CRT,CanRomTao06,DonohoPol}. In \cite{CRT,CanRomTao06,DonohoPol} it was rigorously shown for the first time that in a statistical and large dimensional context a linear sparsity can be recovered from an under-determined system via a simple polynomial $\ell_1$-optimization algorithm. \cite{CanRomTao06} went even further and established that in \emph{noisy} systems for any linear level of under-determinedness there is again a linear sparsity that can be \emph{approximately} recovered through an SOCP (second order cone programming) noisy equivalent to $\ell_1$. Moreover, the approximate solution is (in an $\ell_2$-norm sense) guaranteed to be no further from the sparse unknown vector than a constant times the noise. In this paper we will also consider solving \emph{noisy} linear systems and present an alternative statistical framework that can be used for their analysis. To demonstrate how the framework works we will show how one can use it to precisely characterize the approximation error of a wide class of SOCP algorithms. We will also show that our theoretical predictions are in a solid agrement with the results one can get through numerical simulations.

cs.IT↗

arXiv2013

Meshes that trap random subspaces

In our recent work \cite{StojnicCSetam09,StojnicUpper10} we considered solving under-determined systems of linear equations with sparse solutions. In a large dimensional and statistical context we proved results related to performance of a polynomial $\ell_1$-optimization technique when used for solving such systems. As one of the tools we used a probabilistic result of Gordon \cite{Gordon88}. In this paper we revisit this classic result in its core form and show how it can be reused to in a sense prove its own optimality.

cs.IT↗

arXiv2013

A problem dependent analysis of SOCP algorithms in noisy compressed sensing

Under-determined systems of linear equations with sparse solutions have been the subject of an extensive research in last several years above all due to results of \cite{CRT,CanRomTao06,DonohoPol}. In this paper we will consider \emph{noisy} under-determined linear systems. In a breakthrough \cite{CanRomTao06} it was established that in \emph{noisy} systems for any linear level of under-determinedness there is a linear sparsity that can be \emph{approximately} recovered through an SOCP (second order cone programming) optimization algorithm so that the approximate solution vector is (in an $\ell_2$-norm sense) guaranteed to be no further from the sparse unknown vector than a constant times the noise. In our recent work \cite{StojnicGenSocp10} we established an alternative framework that can be used for statistical performance analysis of the SOCP algorithms. To demonstrate how the framework works we then showed in \cite{StojnicGenSocp10} how one can use it to precisely characterize the \emph{generic} (worst-case) performance of the SOCP. In this paper we present a different set of results that can be obtained through the framework of \cite{StojnicGenSocp10}. The results will relate to \emph{problem dependent} performance analysis of SOCP's. We will consider specific types of unknown sparse vectors and characterize the SOCP performance when used for recovery of such vectors. We will also show that our theoretical predictions are in a solid agreement with the results one can get through numerical simulations.

cs.IT↗

arXiv2009

Various thresholds for $\ell_1$-optimization in compressed sensing

Recently, \cite{CRT,DonohoPol} theoretically analyzed the success of a polynomial $\ell_1$-optimization algorithm in solving an under-determined system of linear equations. In a large dimensional and statistical context \cite{CRT,DonohoPol} proved that if the number of equations (measurements in the compressed sensing terminology) in the system is proportional to the length of the unknown vector then there is a sparsity (number of non-zero elements of the unknown vector) also proportional to the length of the unknown vector such that $\ell_1$-optimization succeeds in solving the system. In this paper, we provide an alternative performance analysis of $\ell_1$-optimization and obtain the proportionality constants that in certain cases match or improve on the best currently known ones from \cite{DonohoPol,DT}.

cs.IT↗

arXiv2009

Block-length dependent thresholds in block-sparse compressed sensing

One of the most basic problems in compressed sensing is solving an under-determined system of linear equations. Although this problem seems rather hard certain $\ell_1$-optimization algorithm appears to be very successful in solving it. The recent work of \cite{CRT,DonohoPol} rigorously proved (in a large dimensional and statistical context) that if the number of equations (measurements in the compressed sensing terminology) in the system is proportional to the length of the unknown vector then there is a sparsity (number of non-zero elements of the unknown vector) also proportional to the length of the unknown vector such that $\ell_1$-optimization algorithm succeeds in solving the system. In more recent papers \cite{StojnicICASSP09block,StojnicJSTSP09} we considered the setup of the so-called \textbf{block}-sparse unknown vectors. In a large dimensional and statistical context, we determined sharp lower bounds on the values of allowable sparsity for any given number (proportional to the length of the unknown vector) of equations such that an $\ell_2/\ell_1$-optimization algorithm succeeds in solving the system. The results established in \cite{StojnicICASSP09block,StojnicJSTSP09} assumed a fairly large block-length of the block-sparse vectors. In this paper we consider the block-length to be a parameter of the system. Consequently, we then establish sharp lower bounds on the values of the allowable block-sparsity as functions of the block-length.

cs.IT↗

arXiv2008

On the reconstruction of block-sparse signals with an optimal number of measurements

Let A be an M by N matrix (M < N) which is an instance of a real random Gaussian ensemble. In compressed sensing we are interested in finding the sparsest solution to the system of equations A x = y for a given y. In general, whenever the sparsity of x is smaller than half the dimension of y then with overwhelming probability over A the sparsest solution is unique and can be found by an exhaustive search over x with an exponential time complexity for any y. The recent work of Candés, Donoho, and Tao shows that minimization of the L_1 norm of x subject to A x = y results in the sparsest solution provided the sparsity of x, say K, is smaller than a certain threshold for a given number of measurements. Specifically, if the dimension of y approaches the dimension of x, the sparsity of x should be K < 0.239 N. Here, we consider the case where x is d-block sparse, i.e., x consists of n = N / d blocks where each block is either a zero vector or a nonzero vector. Instead of L_1-norm relaxation, we consider the following relaxation min x \| X_1 \|_2 + \| X_2 \|_2 + ... + \| X_n \|_2, subject to A x = y where X_i = (x_{(i-1)d+1}, x_{(i-1)d+2}, ..., x_{i d}) for i = 1,2, ..., N. Our main result is that as n -> \infty, the minimization finds the sparsest solution to Ax = y, with overwhelming probability in A, for any x whose block sparsity is k/n < 1/2 - O(ε), provided M/N > 1 - 1/d, and d = Ω(\log(1/ε)/ε). The relaxation can be solved in polynomial time using semi-definite programming.

cs.IT↗

arXiv2003

Optimal quantum detectors for unambiguous detection of mixed states

We consider the problem of designing an optimal quantum detector that distinguishes unambiguously between a collection of mixed quantum states. Using arguments of duality in vector space optimization, we derive necessary and sufficient conditions for an optimal measurement that maximizes the probability of correct detection. We show that the previous optimal measurements that were derived for certain special cases satisfy these optimality conditions. We then consider state sets with strong symmetry properties, and show that the optimal measurement operators for distinguishing between these states share the same symmetries, and can be computed very efficiently by solving a reduced size semidefinite program.

quant-ph↗