Detection limits in the high-dimensional spiked rectangular model

Ahmed El Alaoui, Michael I. Jordan
Proceedings of the 31st Conference On Learning Theory, PMLR 75:410-438, 2018.

Abstract

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 operates through the principle 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, not necessarily given by BBP, 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).

Cite this Paper

BibTeX
@InProceedings{pmlr-v75-el-alaoui18a, title = {Detection limits in the high-dimensional spiked rectangular model}, author = {{El Alaoui}, Ahmed and Jordan, Michael I.}, booktitle = {Proceedings of the 31st Conference On Learning Theory}, pages = {410--438}, year = {2018}, editor = {Bubeck, Sébastien and Perchet, Vianney and Rigollet, Philippe}, volume = {75}, series = {Proceedings of Machine Learning Research}, month = {06--09 Jul}, publisher = {PMLR}, pdf = {http://proceedings.mlr.press/v75/el-alaoui18a/el-alaoui18a.pdf}, url = {https://proceedings.mlr.press/v75/el-alaoui18a.html}, abstract = {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 operates through the principle 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, not necessarily given by BBP, 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). } }
Endnote
%0 Conference Paper %T Detection limits in the high-dimensional spiked rectangular model %A Ahmed El Alaoui %A Michael I. Jordan %B Proceedings of the 31st Conference On Learning Theory %C Proceedings of Machine Learning Research %D 2018 %E Sébastien Bubeck %E Vianney Perchet %E Philippe Rigollet %F pmlr-v75-el-alaoui18a %I PMLR %P 410--438 %U https://proceedings.mlr.press/v75/el-alaoui18a.html %V 75 %X 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 operates through the principle 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, not necessarily given by BBP, 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).
APA
El Alaoui, A. & Jordan, M.I.. (2018). Detection limits in the high-dimensional spiked rectangular model. Proceedings of the 31st Conference On Learning Theory, in Proceedings of Machine Learning Research 75:410-438 Available from https://proceedings.mlr.press/v75/el-alaoui18a.html.

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