Recovering Jointly Sparse Signals via Joint Basis Pursuit

TL;DR

This paper introduces a convex optimization method for recovering signals sparse in two bases simultaneously, demonstrating improved measurement efficiency over traditional single-basis approaches through theoretical analysis and simulations.

Contribution

It develops novel optimality conditions for joint basis pursuit and shows that fewer measurements are needed for successful recovery compared to classical methods.

Findings

Requires as few as O(max{k1,k2} log log n) measurements for recovery

Outperforms traditional methods requiring Θ(min{k1,k2} log(n/min{k1,k2})) measurements

Analysis is supported by extensive simulations indicating tight bounds

Abstract

This work considers recovery of signals that are sparse over two bases. For instance, a signal might be sparse in both time and frequency, or a matrix can be low rank and sparse simultaneously. To facilitate recovery, we consider minimizing the sum of the $\ell_1$-norms that correspond to each basis, which is a tractable convex approach. We find novel optimality conditions which indicates a gain over traditional approaches where $\ell_1$ minimization is done over only one basis. Next, we analyze these optimality conditions for the particular case of time-frequency bases. Denoting sparsity in the first and second bases by $k_1,k_2$ respectively, we show that, for a general class of signals, using this approach, one requires as small as $O(\max\{k_1,k_2\}\log\log n)$ measurements for successful recovery hence overcoming the classical requirement of $\Theta(\min\{k_1,k_2\}\log(\frac{n}{\min\{k_1,k_2\}}))$ for $\ell_1$ minimization when $k_1\approx k_2$. Extensive simulations show that, our analysis is approximately tight.