A Sequential Approach for Sparse Support Recovery using Correlation Priors

Abstract

Recovering the support of a sparse vector from its linear underdetermined noisy measurements is a fundamental problem in signal processing and statistics. In many applications, one has the additional ability to design the rows of measurement matrix based on already collected measurements, giving rise to concept of adaptive support recovery. Compared to traditional batch support recovery, adaptive settings generally have a better performance in presence of noise, and have lower computational complexity. When only a single measurement vector is available, it has been shown that both in adaptive and batch settings, one cannot recover supports of size k>M, where M is the size of measurement vector. However, in presence of L> 1 measurement vectors, and certain additional statistical assumptions, we have shown that one can recover supports of size $k = \mathcal{O}\left( {{M^2}} \right)$, when batch measurements are available. This paper considers the unexplored case where we have both the abilities of collecting multiple measurement vectors, and adaptively designing the measurement matrix. We propose an adaptive support recovery algorithm with a low computational cost, and provide non-asymptotic guarantees for our proposed algorithm which show that probability of detecting a wrong support goes to zero exponentially fast as the number of measurement vectors goes to infinity1.