Multiple Support Recovery Using Very Few Measurements Per Sample

TL;DR

This paper introduces a novel method for recovering multiple sparse supports from very few measurements per sample, using a two-step spectral approach that works even when measurements are fewer than the support size.

Contribution

The paper proposes a new two-step spectral algorithm for support recovery in a measurement-constrained regime, enabling support identification with fewer measurements than support size.

Findings

Supports can be recovered with fewer than support size measurements per sample.

The method works under general generative models for samples and measurement matrices.

Experimental results validate the effectiveness on synthetic and MNIST data.

Abstract

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.