Improved analysis of greedy block coordinate descent under RIP

Abstract

A more relaxed condition means that fewer of measurements are needed to ensure the exact sparse recovery from the theoretical aspect. The sufficient condition for the greedy block coordinate descent (GBCD) algorithm is relaxed using the near-orthogonality property. It is also shown that the GBCD algorithm fails when (1/(√K+1)≤δ K+1 <1).

Sufficient condition for exact support recovery of sparse signals through greedy block coordinate descent

“…By using the fact that the MMV problem can be viewed as block-sparse recovery and the property of norm, through constructing perfect square expression, we improve the results proposed in [7]. Recently, based on the near orthogonality property proposed by Chang and Wu [8], Li et al [9] improved the theoretical guarantee for the GBCD algorithm. In the noiseless case, the sufficient condition proposed by Li et al [7] was improved to [9] also claimed that the GBCD fails in a special case when 1/ K + 1 ≤ δ K + 1 < 1.…”

“…Recently, based on the near orthogonality property proposed by Chang and Wu [8], Li et al [9] improved the theoretical guarantee for the GBCD algorithm. In the noiseless case, the sufficient condition proposed by Li et al [7] was improved to [9] also claimed that the GBCD fails in a special case when 1/ K + 1 ≤ δ K + 1 < 1. In [10], in the noiseless case, the authors showed that if A satisfies the RIP with δ K + 1 < (1/( K + 1)), then GBCD recovers the support of any K-group sparse matrix X in K iterations.…”

“…In addition, the authors of [18,19] used this fact to address the magnetic resonance image reconstruction problem. In this study, we use this fact to improve the sufficient condition proposed by the authors of [7,9] and we present an optimal condition for exact support recovery with GBCD in the noiseless case. We also show that the GBCD fails in a more general case.…”

“…then the GBCD algorithm can exactly recover the support of X. Remark 3 (comparison with [9]): In [9], it was claimed that if A satisfies the RIP of order K + 1 with…”

Support Recovery of Greedy Block Coordinate Descent Using the Near Orthogonality Property

“…As cited in [3], the near orthogonality property can further develop the orthogonality characterization of columns in A; it will play a fundamental role in the study of the signal reconstruction performance in compressed sensing. In the noiseless case, the work of [15] analyzed the performance of GBCD using near orthogonality property and improved the results in [2].…”