A Sharp Condition for Exact Support Recovery of Sparse Signals With Orthogonal Matching Pursuit

Support recovery of sparse signals from noisy measurements with orthogonal matching pursuit (OMP) has been extensively studied in the literature. In this paper, we show that for any $K$-sparse signal \x, if the sensing matrix \A satisfies the restricted isometry property (RIP) of order $K + 1$ with restricted isometry constant (RIC) $\delta_{K+1} < 1/\sqrt {K+1}$, then under some constraint on the minimum magnitude of the nonzero elements of \x, the OMP algorithm exactly recovers the support of \x from the measurements \y=\A\x+\v in $K$ iterations, where \v is the noise vector. This condition is sharp in terms of $\delta_{K+1}$ since for any given positive integer $K\geq 2$ and any $1/\sqrt{K+1}\leq t<1$, there always exist a $K$-sparse \x and a matrix \A satisfying $\delta_{K+1}=t$ for which OMP may fail to recover the signal \x in $K$ iterations. Moreover, the constraint on the minimum magnitude of the nonzero elements of \x is weaker than existing results.Comment: ISIT 2016, 2364-2368. arXiv admin note: text overlap with arXiv:1512.07248

A sharp recovery condition for sparse signals with partial support information via orthogonal matching pursuit

This paper considers the exact recovery of $k$-sparse signals in the noiseless setting and support recovery in the noisy case when some prior information on the support of the signals is available. This prior support consists of two parts. One part is a subset of the true support and another part is outside of the true support. For $k$-sparse signals $\mathbf{x}$ with the prior support which is composed of $g$ true indices and $b$ wrong indices, we show that if the restricted isometry constant (RIC) $\delta_{k+b+1}$ of the sensing matrix $\mathbf{A}$ satisfies \begin{eqnarray*} \delta_{k+b+1}<\frac{1}{\sqrt{k-g+1}}, \end{eqnarray*} then orthogonal matching pursuit (OMP) algorithm can perfectly recover the signals $\mathbf{x}$ from $\mathbf{y}=\mathbf{Ax}$ in $k-g$ iterations. Moreover, we show the above sufficient condition on the RIC is sharp. In the noisy case, we achieve the exact recovery of the remainder support (the part of the true support outside of the prior support) for the $k$-sparse signals $\mathbf{x}$ from $\mathbf{y}=\mathbf{Ax}+\mathbf{v}$ under appropriate conditions. For the remainder support recovery, we also obtain a necessary condition based on the minimum magnitude of partial nonzero elements of the signals $\mathbf{x}$

A Sharp Condition for Exact Support Recovery of with Orthogonal Matching Pursuit

Support recovery of sparse signals from noisy measurements with orthogonal matching pursuit (OMP) has been extensively studied. In this paper, we show that for any $K$-sparse signal \x, if a sensing matrix \A satisfies the restricted isometry property (RIP) with restricted isometry constant (RIC) $\delta_{K+1} < 1/\sqrt {K+1}$, then under some constraints on the minimum magnitude of nonzero elements of \x, OMP exactly recovers the support of \x from its measurements \y=\A\x+\v in $K$ iterations, where \v is a noise vector that is $\ell_2$ or $\ell_{\infty}$ bounded. This sufficient condition is sharp in terms of $\delta_{K+1}$ since for any given positive integer $K$ and any $1/\sqrt{K+1}\leq \delta<1$, there always exists a matrix \A satisfying the RIP with $\delta_{K+1}=\delta$ for which OMP fails to recover a $K$-sparse signal \x in $K$ iterations. Also, our constraints on the minimum magnitude of nonzero elements of \x are weaker than existing ones. Moreover, we propose worst-case necessary conditions for the exact support recovery of \x, characterized by the minimum magnitude of the nonzero elements of \x.Comment: Jinming Wen, Zhengchun Zhou, Jian Wang, Xiaohu, Tang and Qun Mo. A Sharp Condition for Exact Support Recovery with Orthogonal Matching Pursuit, IEEE Transactions on Signal Processing, 65(2017),1370-138

Signal and Noise Statistics Oblivious Sparse Reconstruction using OMP/OLS

Orthogonal matching pursuit (OMP) and orthogonal least squares (OLS) are widely used for sparse signal reconstruction in under-determined linear regression problems. The performance of these compressed sensing (CS) algorithms depends crucially on the \textit{a priori} knowledge of either the sparsity of the signal ($k_0$) or noise variance ($\sigma^2$). Both $k_0$ and $\sigma^2$ are unknown in general and extremely difficult to estimate in under determined models. This limits the application of OMP and OLS in many practical situations. In this article, we develop two computationally efficient frameworks namely TF-IGP and RRT-IGP for using OMP and OLS even when $k_0$ and $\sigma^2$ are unavailable. Both TF-IGP and RRT-IGP are analytically shown to accomplish successful sparse recovery under the same set of restricted isometry conditions on the design matrix required for OMP/OLS with \textit{a priori} knowledge of $k_0$ and $\sigma^2$. Numerical simulations also indicate a highly competitive performance of TF-IGP and RRT-IGP in comparison to OMP/OLS with \textit{a priori} knowledge of $k_0$ and $\sigma^2$.Comment: 14 pages and 7 figure

Exact Support and Vector Recovery of Constrained Sparse Vectors via Constrained Matching Pursuit

Matching pursuit, especially its orthogonal version (OMP) and variations, is a greedy algorithm widely used in signal processing, compressed sensing, and sparse modeling. Inspired by constrained sparse signal recovery, this paper proposes a constrained matching pursuit algorithm and develops conditions for exact support and vector recovery on constraint sets via this algorithm. We show that exact recovery via constrained matching pursuit not only depends on a measurement matrix but also critically relies on a constraint set. We thus identify an important class of constraint sets, called coordinate projection admissible set, or simply CP admissible sets; analytic and geometric properties of these sets are established. We study exact vector recovery on convex, CP admissible cones for a fixed support. We provide sufficient exact recovery conditions for a general support as well as necessary and sufficient recovery conditions when a support has small size. As a byproduct, we construct a nontrivial counterexample to a renowned necessary condition of exact recovery via the OMP for a support of size three. Moreover, using the properties of convex CP admissible sets and convex optimization techniques, we establish sufficient conditions for uniform exact recovery on convex CP admissible sets in terms of the restricted isometry-like constant and the restricted orthogonality-like constant