Compressive MUSIC: Revisiting the Link Between Compressive Sensing and Array Signal Processing

Abstract

The multiple measurement vector (MMV) problem addresses the identification of unknown input vectors that share common sparse support. Even though MMV problems have been traditionally addressed within the context of sensor array signal processing, the recent trend is to apply compressive sensing (CS) due to its capability to estimate sparse support even with an insufficient number of snapshots, in which case classical array signal processing fails. However, CS guarantees the accurate recovery in a probabilistic manner, which often shows inferior performance in the regime where the traditional array signal processing approaches succeed. The apparent dichotomy between the probabilistic CS and deterministic sensor array signal processing has not been fully understood. The main contribution of the present article is a unified approach that unveils a missing link between CS and array signal processing. The new algorithm, which we call compressive MUSIC, identifies the parts of support using CS, after which the remaining supports are estimated using a novel generalized MUSIC criterion. Using a large system MMV model, we show that our compressive MUSIC requires a smaller number of sensor elements for accurate support recovery than the existing CS methods and that it can approach the optimal l0-bound with finite number of snapshots. multiple measurement vector (MMV) problem is formulated as:The MMV problem also has many important applications such as distributed compressive sensing [25], direction-of-arrival estimation in radar [26], magnetic resonance imaging with multiple coils [27], diffuse optical tomography using multiple illumination patterns [28, 29], etc. Currently, greedy algorithms such as S-OMP (simultaneous orthogonal matching pursuit) [21, 30], convex relaxation methods using mixed norm [31, 32], M-FOCUSS [22], M-SBL (Multiple Sparse Bayesian Learning) [33], randomized algorithms such as REduce MMV and BOost (ReMBo)[23], and model-based compressive sensing using block-sparsity [34, 35] have also been applied to the MMV problem within the context of compressive sensing.In MMV, thanks to the common sparse support, it is quite predictable that the recoverable sparsity level may increase with the increasing number of measurement vectors. More specifically, given a sensing matrix A, let spark(A) denote the smallest number of linearly dependent columns of A. Then, according to Chen and Huo [21], Feng and Bresler [36], if X ∈ R n×r satisfies AX = B and X 0 < spark(A) + rank(B) − 1 2 ≤ spark(A) − 1, (I.3) then X is the unique solution of (I.2). In (I.3), the last inequality comes from the observation that rank(B) ≤ X * 0 := |suppX * |. Recently, Davies and Eldar showed that (I.3) is indeed a necessary codition for X to be a unique solution for AX = B [37]. Compared to the SMV case (rank(B) = 1), (I.3) informs us that the recoverable sparsity level increases with the number of measurement vectors. Furthermore, average case analysis [38] and information theoretic analysis [39] have indicated the performance improvement...

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“…Many algorithms were proposed to solve the MMV problem, such as Multiple Signal Classification algorithms [10][11][12][13], Bayesian learning algorithms [14][15][16], and greedy pursuit (GP) algorithms [17][18][19][20] as well as 1 − SVD [6] and − thresholding algorithm [21]. Generally, GP algorithms are easily implemented and have low computational complexity.…”

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