Exploiting Prior Information in Block-Sparse Signals

Abstract

We study the problem of recovering a block-sparse signal from under-sampled observations. The non-zero values of such signals appear in few blocks, and their recovery is often accomplished using an 1,2 optimization problem. In applications such as DNA micro-arrays, some extra information about the distribution of non-zero blocks is available; i.e., the number of non-zero blocks in certain subsets of the blocks is known. A typical way to consider the extra information in recovery procedures is to solve a weighted 1,2 problem. In this paper, we consider a block-sparse model which is accompanied with a partitioning of the blocks; besides the overall block-sparsity level of the signal, we assume to know the block-sparsity of each subset in the partition. Our goal in this work is to minimize the number of required linear measurements for perfect recovery of the signal by tuning the weights of a weighted 1,2 problem. For this goal, we apply tools from conic integral geometry and derive closedform expressions for the optimal weights. We show through precise analysis and simulations that the weighted 1,2 problem with optimal weights significantly outperforms the regular 1,2 problem. We further show that the optimal weights are robust against the inaccuracies of prior information.Index Terms-block-sparse, prior information, weighted 1,2, conic integral geometry.

Improved Recovery of Analysis Sparse Vectors in Presence of Prior Information

“…In practice, one may encounter some inaccuracies in determining α P R L . The study of the sensitivity of weights to the inaccuracies in α were previously considered in [10]. Fortunately, small changes in α are shown to have insignificant impact on the derived weights.…”

Distribution-Aware Block-Sparse Recovery via Convex Optimization

“…CS in presence of prior information has been studied in different signal models. While a large part of research (see for example [12]- [17]) deals with deterministic signal models, only a few works (see [18]- [21]) have investigated random signal models with Bayesian information. In the deterministic model, the ground-truth signal has intersected with a few sets which called support estimates.…”

“…The contributing level of each set to the support is available to the experimenter [12], [13]. This exact situation is investigated in [17]. They propose a non-uniform model for capturing deterministic prior information.…”

“…We proceed by using a closed-form expression for δpDpB, wqq a special case of which is obtained in [17,Lemma 2].…”

“…Moreover, the minimum is achieved at a unique t ě 0. The inequality is in fact equality in the asymptotic case (q Ñ 8) [17,Proposition 3].…”