Detection limits in the high-dimensional spiked rectangular model
This work addresses detection limits in high-dimensional statistics, providing theoretical insights for signal processing and machine learning, but it is incremental as it parallels earlier findings for spherical spikes.
The paper tackles the problem of detecting a single unknown spike in a high-dimensional rectangular data matrix, showing that the likelihood ratio has asymptotically Gaussian fluctuations below the BBP threshold, with results extending to generic product priors including sparsity.
We study the problem of detecting the presence of a single unknown spike in a rectangular data matrix, in a high-dimensional 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 so-called BBP threshold from random matrix theory. Our result parallels earlier findings of Onatski et al.\ (2013) and Johnstone-Onatski (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 is based on Talagrand's interpretation of the cavity method from spin-glass theory. The question of the maximal parameter region where asymptotic normality is expected to hold is left open. This region is shaped by the prior in a non-trivial way. We conjecture that this is the entire paramagnetic phase of an associated spin-glass model, and is defined by the vanishing of the replica-symmetric solution of Lesieur et al.\ (2015).