Batch Sparse Recovery, or How to Leverage the Average Sparsity

Abstract

We introduce a \emph{batch} version of sparse recovery, where the goal is to report a sequence of vectors $A_1^{\prime },\dots ,A_m^{\prime }\in {\mathbb{R}}^n$ that estimate unknown signals $A_1,\dots ,A_m\in {\mathbb{R}}^n$ using a few linear measurements, each involving exactly one signal vector, under an assumption of \emph{average sparsity}. More precisely, we want to have \newline $(1){\sum }_{j\in [m]}\Vert A_j-A_j^{\prime }{\Vert }_p^p\le C\cdot \min \{{\sum }_{j\in [m]}\Vert A_j-A_j^{\ast }{\Vert }_p^p\}$ for predetermined constants $C\ge 1$ and $p$, where the minimum is over all $A_1^{\ast },\dots ,A_m^{\ast }\in {\mathbb{R}}^n$ that are $k$-sparse on average. We assume $k$ is given as input, and ask for the minimal number of measurements required to satisfy $(1)$. The special case $m=1$ is known as stable sparse recovery and has been studied extensively. We resolve the question for $p=1$ up to polylogarithmic factors, by presenting a randomized adaptive scheme that performs $\tilde{O}(km)$ measurements and with high probability has output satisfying $(1)$, for arbitrarily small $C>1$. Finally, we show that adaptivity is necessary for every non-trivial scheme.