Sparse recovery with coherent tight frames via analysis Dantzig selector and analysis LASSO

Abstract

This article considers recovery of signals that are sparse or approximately sparse in terms of a (possibly) highly overcomplete and coherent tight frame from undersampled data corrupted with additive noise. We show that the properly constrained l 1 -analysis optimization problem, called analysis Dantzig selector, stably recovers a signal which is nearly sparse in terms of a tight frame provided that the measurement matrix satisfies a restricted isometry property adapted to the tight frame. As a special case, we consider the Gaussian noise. Further, under a sparsity scenario, with high probability, the recovery error from noisy data is within a log-like factor of the minimax risk over the class of vectors which are at most s sparse in terms of the tight frame. Similar results for the analysis LASSO are shown.The above two algorithms provide guarantees only for noise that is bounded or bounded with high probability (for example, Gaussian noise). However, when the underlying measurements are corrupted by sparse noise, these algorithms perform suboptimally. We demonstrate robust methods for reconstructing signals that are nearly sparse in terms of a tight frame in the presence of bounded noise combined with sparse noise. The analysis in this paper is based on the restricted isometry property adapted to a tight frame, which is a natural extension to the standard restricted isometry property.

Smoothing and Decomposition for Analysis Sparse Recovery

“…As a special case, choosing ρ → ∞ extends the bound in (6) and obtains the reconstruction bound for ALASSO (4) as long as σ 2s < 0.1907, which improves upon the results of [28].…”

“…A similar performance bound is introduced in [28] and shown to be valid when σ 3s < 0.25. Using Corollary 3.4 in [35], this is equivalent to σ 2s < 0.0833.…”

“…The proof follows mainly from the ideas in [9], [28], and proceeds in two steps. First, we try to show that D * h inside T 01 is bounded by the terms of D * h outside the set T .…”

“…With the introduction of the restricted isometry property adapted to D (D-RIP) [12], previous work [12] [28] studied recovery guarantees based on ABP (3) and ALASSO (4). Here we combine the techniques in [9] and [28], and obtain a performance bound on RALASSO (5). We show that when σ 2s < 0.1907…”

Stable recovery of analysis based approaches

“…Some sufficient conditions are provided to guarantee the stable recovery via solving analysis based approaches. Compared with the previous work [12,16], our sufficient conditions are weaker and the estimations of l 2 bound only depend on the measurement matrix.…”