FUNDAMENTAL LIMITS OF DETECTION IN THE SPIKED WIGNER MODEL

• 成果类型: Article
• 署名作者: El Alaoui, Ahmed; Krzakala, Florent; Jordan, Michael
• 署名单位: Stanford University; Sorbonne Universite; Universite PSL; Ecole Normale Superieure (ENS); Universite Paris Cite; Centre National de la Recherche Scientifique (CNRS); Sorbonne Universite; University of California System; University of California Berkeley
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/19-AOS1826
• 发表日期: 2020
• 页码: 863-885
• 关键词: largest eigenvalue; principal-components; free-energy; matrix; fluctuations; deformation; dimension; tests; PCA

摘要

We study the fundamental limits of detecting the presence of an additive rank-one perturbation, or spike, to a Wigner matrix. When the spike comes from a prior that is i.i.d. across coordinates, we prove that the log-likelihood ratio of the spiked model against the nonspiked one is asymptotically normal below a certain reconstruction threshold which is not necessarily of a spectral nature, and that it is degenerate above. This establishes the maximal region of contiguity between the planted and null models. It is known that this threshold also marks a phase transition for estimating the spike: the latter task is possible above the threshold and impossible below. Therefore, both estimation and detection undergo the same transition in this random matrix model. Further information on the performance of the optimal test is also provided. Our proofs are based on Gaussian interpolation methods and a rigorous incarnation of the cavity method, as devised by Guerra and Talagrand in their study of the Sherrington-Kirkpatrick spin-glass model.