Related papers: Upper-bounding $\ell_1$-optimization weak threshol…
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…
In this paper we look at a particular problem related to under-determined linear systems of equations with sparse solutions. $\ell_1$-minimization is a fairly successful polynomial technique that can in certain statistical scenarios find…
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…
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}…
In this paper we consider under-determined systems of linear equations that have sparse solutions. This subject attracted enormous amount of interest in recent years primarily due to influential works \cite{CRT,DonohoPol}. In a statistical…
In this paper we revisit under-determined linear systems of equations with sparse solutions. As is well known, these systems are among core mathematical problems of a very popular compressed sensing field. The popularity of the field as…
Recent studies of under-determined linear systems of equations with sparse solutions showed a great practical and theoretical efficiency of a particular technique called $\ell_1$-optimization. Seminal works \cite{CRT,DOnoho06CS} rigorously…
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…
In this paper we look at a connection between the $\ell_q,0\leq q\leq 1$, optimization and under-determined linear systems of equations with sparse solutions. The case $q=1$, or in other words $\ell_1$ optimization and its a connection with…
In this paper we revisit one of the classical problems of compressed sensing. Namely, we consider linear under-determined systems with sparse solutions. A substantial success in mathematical characterization of an $\ell_1$ optimization…
In this paper we revisit random linear under-determined systems with sparse solutions. We consider $\ell_1$ optimization heuristic known to work very well when used to solve these systems. A collection of fundamental results that relate to…
In this paper we consider solving \emph{noisy} under-determined systems of linear equations with sparse solutions. A noiseless equivalent attracted enormous attention in recent years, above all, due to work of…
In this paper we look at a well known linear inverse problem that is one of the mathematical cornerstones of the compressed sensing field. In seminal works \cite{CRT,DOnoho06CS} $\ell_1$ optimization and its success when used for recovering…
Motivated by problems in optimization we study the sparsity of the solutions to systems of linear Diophantine equations and linear integer programs, i.e., the number of non-zero entries of a solution, which is often referred to as the…
We consider the problem of recovering a partially sparse solution of an underdetermined system of linear equations by minimizing the $\ell_1$-norm of the part of the solution vector which is known to be sparse. Such a problem is closely…
Sparse estimation methods are aimed at using or obtaining parsimonious representations of data or models. While naturally cast as a combinatorial optimization problem, variable or feature selection admits a convex relaxation through the…
In this paper we consider random linear under-determined systems with block-sparse solutions. A standard subvariant of such systems, namely, precisely the same type of systems without additional block structuring requirement, gained a lot…
There is a recent surge of interest in developing algorithms for finding sparse solutions of underdetermined systems of linear equations $y = \Phi x$. In many applications, extremely large problem sizes are envisioned, with at least tens of…
For statistical modeling wherein the data regime is unfavorable in terms of dimensionality relative to the sample size, finding hidden sparsity in the ground truth can be critical in formulating an accurate statistical model. The so-called…
Recovery of the sparsity pattern (or support) of an unknown sparse vector from a small number of noisy linear measurements is an important problem in compressed sensing. In this paper, the high-dimensional setting is considered. It is shown…