Detection limits in the highdimensional spiked rectangular model
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Proceedings of the 31st Conference On Learning Theory, PMLR 75:410438, 2018.
Abstract
We study the problem of detecting the presence of a single unknown spike in a rectangular data matrix, in a highdimensional regime where the spike has fixed strength and the aspect ratio of the matrix converges to a finite limit. This setup includes Johnstone’s spiked covariance model. We analyze the likelihood ratio of the spiked model against an “all noise" null model of reference, and show it has asymptotically Gaussian fluctuations in a region below—but in general not up to—the socalled BBP threshold from random matrix theory. Our result parallels earlier findings of Onatski et al. (2013) and JohnstoneOnatski (2015) for spherical spikes. We present a probabilistic approach capable of treating generic product priors. In particular, sparsity in the spike is allowed. Our approach operates through the principle of the cavity method from spinglass theory. The question of the maximal parameter region where asymptotic normality is expected to hold is left open. This region, not necessarily given by BBP, is shaped by the prior in a nontrivial way. We conjecture that this is the entire paramagnetic phase of an associated spinglass model, and is defined by the vanishing of the replicasymmetric solution of Lesieur et al. (2015).
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