Sample-Measurement Tradeoff in Support Recovery under a Subgaussian Prior
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Data samples from R^d with a common support of size k are accessed through m random linear projections per sample. It is well-known that roughly k measurements from a single sample are sufficient to recover the support. In the multiple sample setting, do k overall measurements still suffice when only m<k measurements per sample are allowed? We answer this question in the negative by considering a generative model setting with independent samples drawn from a subgaussian prior. We show that n=Θ((k^2/m^2)·log k(d-k)) samples are necessary and sufficient to recover the support exactly. In turn, this shows that k overall samples are insufficient for support recovery when m<k; instead we need about k^2/m overall measurements. Our proposed sample-optimal estimator has a closed-form expression, has computational complexity of O(d nm), and is of independent interest.
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