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 ℝd. These samples can be partitioned into ℓ 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 ℓ 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 tildeO(k4ℓ4/m4) 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.