On sharp performance bounds for robust sparse signal recoveries

It is well known in compressive sensing that l_1 minimization can recover the sparsest solution for a large class of underdetermined systems of linear equations, provided the signal is sufficiently sparse. In this paper, we compute sharp performance bounds for several different notions of robustness in sparse signal recovery via l_1 minimization. In particular, we determine necessary and sufficient conditions for the measurement matrix A under which l_1 minimization guarantees the robustness of sparse signal recovery in the "weak", "sectional" and "strong" (e.g., robustness for "almost all" approximately sparse signals, or instead for "all" approximately sparse signals). Based on these characterizations, we are able to compute sharp performance bounds on the tradeoff between signal sparsity and signal recovery robustness in these various senses. Our results are based on a high-dimensional geometrical analysis of the null-space of the measurement matrix A. These results generalize the thresholds results for purely sparse signals and also present generalized insights on l_1 minimization for recovering purely sparse signals from a null-space perspective

Compressed Sensing over the Grassmann Manifold: A Unified Analytical Framework

It is well known that compressed sensing problems reduce to finding the sparse solutions for large under-determined systems of equations. Although finding the sparse solutions in general may be computationally difficult, starting with the seminal work of [2], it has been shown that linear programming techniques, obtained from an l_(1)-norm relaxation of the original non-convex problem, can provably find the unknown vector in certain instances. In particular, using a certain restricted isometry property, [2] shows that for measurement matrices chosen from a random Gaussian ensemble, l_1 optimization can find the correct solution with overwhelming probability even when the support size of the unknown vector is proportional to its dimension. The paper [1] uses results on neighborly polytopes from [6] to give a ldquosharprdquo bound on what this proportionality should be in the Gaussian measurement ensemble. In this paper we shall focus on finding sharp bounds on the recovery of ldquoapproximately sparserdquo signals (also possibly under noisy measurements). While the restricted isometry property can be used to study the recovery of approximately sparse signals (and also in the presence of noisy measurements), the obtained bounds can be quite loose. On the other hand, the neighborly polytopes technique which yields sharp bounds for ideally sparse signals cannot be generalized to approximately sparse signals. In this paper, starting from a necessary and sufficient condition for achieving a certain signal recovery accuracy, using high-dimensional geometry, we give a unified null-space Grassmannian angle-based analytical framework for compressive sensing. This new framework gives sharp quantitative tradeoffs between the signal sparsity and the recovery accuracy of the l_1 optimization for approximately sparse signals. As it will turn out, the neighborly polytopes result of [1] for ideally sparse signals can be viewed as a special case of ours. Our result concerns fundamental properties of linear subspaces and so may be of independent mathematical interest

A Rice method proof of the Null-Space Property over the Grassmannian

The Null-Space Property (NSP) is a necessary and sufficient condition for the recovery of the largest coefficients of solutions to an under-determined system of linear equations. Interestingly, this property governs also the success and the failure of recent developments in high-dimensional statistics, signal processing, error-correcting codes and the theory of polytopes. Although this property is the keystone of $\ell\_{1}$-minimization techniques, it is an open problem to derive a closed form for the phase transition on NSP. In this article, we provide the first proof of NSP using random processes theory and the Rice method. As a matter of fact, our analysis gives non-asymptotic bounds for NSP with respect to unitarily invariant distributions. Furthermore, we derive a simple sufficient condition for NSP.Comment: 18 Pages, some Figure