Phase Transition for Support Recovery from Gaussian Linear Measurements
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We study the problem of recovering the common k-sized support of a set of n samples of dimension d, using m noisy linear measurements per sample. Most prior work has focused on the case when m exceeds k, in which case n of the order (k/m)log(d/k) is both necessary and sufficient. Thus, in this regime, only the total number of measurements across the samples matter, and there is not much benefit in getting more than k measurements per sample. In the measurement-constrained regime where we have access to fewer than k measurements per sample, we show an upper bound of O((k^2/m^2)log d) on the sample complexity for successful support recovery when m≥ 2log d. Along with the lower bound from our previous work, this shows a sharp phase transition for the sample complexity of this problem around k/m=1. In fact, our proposed algorithm is sample-optimal in both the regimes. It follows that, in the m≪ k regime, multiple measurements from the same sample are more valuable than measurements from different samples.
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