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.