Multiple Support Recovery Using Very Few Measurements Per Sample

In the problem of multiple support recovery, we are given access to linear measurements of multiple sparse samples in $\mathbb{R}^{d}$. These samples can be partitioned into $\ell$ groups, with samples having the same support belonging to the same group. For a given budget of $m$ measurements per sample, the goal is to recover the $\ell$ underlying supports, in the absence of the knowledge of group labels. We study this problem with a focus on the measurement-constrained regime where $m$ is smaller than the support size $k$ of each sample. We design a two-step procedure that estimates the union of the underlying supports first, and then uses a spectral algorithm to estimate the individual supports. Our proposed estimator can recover the supports with $m<k$ measurements per sample, from $\tilde{O}(k^{4}\ell^{4}/m^{4})$ samples. Our guarantees hold for a general, generative model assumption on the samples and measurement matrices. We also provide results from experiments conducted on synthetic data and on the MNIST dataset.