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In the problem of multiple support recovery, we are given access to linear measurements of multiple sparse samples in ℝ^d. 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 Õ(k^4ℓ^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.