Exploiting Prior Information in Block-Sparse Signals

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 <inline-formula><tex-math notation="LaTeX">$\ell _{1,2}$</tex-math></inline-formula> 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 <inline-formula><tex-math notation="LaTeX">$\ell _{1,2}$</tex-math></inline-formula> 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 <inline-formula><tex-math notation="LaTeX">$\ell _{1,2}$</tex-math></inline-formula> problem. For this goal, we apply tools from conic integral geometry and derive closed-form expressions for the optimal weights. We show through precise analysis and simulations that the weighted <inline-formula><tex-math notation="LaTeX">$\ell _{1,2}$</tex-math></inline-formula> problem with optimal weights significantly outperforms the regular <inline-formula><tex-math notation="LaTeX">$\ell _{1,2}$</tex-math></inline-formula> problem. We further show that the optimal weights are robust against the inaccuracies of prior information.