Subspace Methods for Joint Sparse Recovery

Abstract

We propose robust and efficient algorithms for the joint sparse recovery problem in compressed sensing, which simultaneously recover the supports of jointly sparse signals from their multiple measurement vectors obtained through a common sensing matrix.In a favorable situation, the unknown matrix, which consists of the jointly sparse signals, has linearly independent nonzero rows.In this case, the MUSIC (MUltiple SIgnal Classification) algorithm, originally proposed by Schmidt for the direction of arrival problem in sensor array processing and later proposed and analyzed for joint sparse recovery by Feng and Bresler, provides a guarantee with the minimum number of measurements. We focus instead on the unfavorable but practically significant case of rank-defect or ill-conditioning. This situation arises with limited number of measurement vectors, or with highly correlated signal components. In this case MUSIC fails, and in practice none of the existing methods can consistently approach the fundamental limit. We propose subspace-augmented MUSIC (SA-MUSIC), which improves on MUSIC so that the support is reliably recovered under such unfavorable conditions. Combined with subspace-based greedy algorithms also proposed and analyzed in this paper, SA-MUSIC provides a computationally efficient algorithm with a performance guarantee. The performance guarantees are given in terms of a version of restricted isometry property. In particular, we also present a non-asymptotic perturbation analysis of the signal subspace estimation that has been missing in the previous study of MUSIC. Index TermsCompressed sensing, joint sparsity, multiple measurement vectors (MMV), subspace estimation, restricted isometry property (RIP), sensor array processing, spectrum-blind sampling.

Improving Noise Robustness in Subspace-Based Joint Sparse Recovery

“…The elements of a sensing matrix A were generated from a Gaussian distribution having zero mean and variance of 1/m, and then each column of A was normalized to have an unit norm. An unknown signal X with rank(X) = r ≤ k was generated using the same procedure as in [12]. Specifically, we randomly generated a support I, and then the corresponding nonzero signal components were obtained by…”

“…In order to identify the contribution of the forward and backward greedy steps in the performance improvement, we perform additional experiments using the same simulation setup. estimated using an identical subspace S-OMP algorithm in [7], [12] so that performance differences came only from the sequential subspace estimation step. For a bigger N where the signal subspace error is small, performance improvement due to the sequential subspace estimation was not remarkable.…”

“…Recently, Kim et al [7] and Lee et al [12] independently showed that such a diversity gain can be also achieved in a new class of greedy algorithms by exploiting the so-called generalized (or extended) multiple signal classification (MUSIC) criterion [7], [12]. More specifically, these algorithms obtain a partial support estimate using a conventional MMV greedy algorithm, and then the atoms corresponding to the partial supports are augmented into a data matrix to obtain an augumented signal subspace estimate.…”

“…The performance improvement of these greedy algorithms is substantial and nearly achieves the l 0 bound when a signal subspace and partial support estimation are accurate due to a sufficient number of snapshots or high signal to noise ratio (SNR) [7], [12]. However, if either of these estimation is erroneous owing to an insufficient number of snapshots or low SNR, performance degrades.…”

“…A central theme in these studies has been that joint sparsity within signal ensembles enables a further reduction in the number of required measurements [5], [6], where the number of measurements required per sensor must account for the minimal features unique to that sensor [7], [8], [9], [10], [11], [12].…”