Pseudo-maximum likelihood theory for high-dimensional rank one inference
Abstract
We develop a pseudo-likelihood theory for rank one matrix estimation problems in the high-dimensional limit. We prove a variational principle for the limiting pseudo-maximum likelihood, which also characterizes the performance of the corresponding pseudo-maximum likelihood estimator. We show that this variational principle is universal and depends only on four parameters determined by the corresponding null model. Through this universality, we introduce a notion of equivalence for estimation problems of this type and, in particular, show that a broad class of estimation tasks, including community detection, sparse submatrix detection and nonlinear spiked matrix models, are equivalent to spiked matrix models. As an application, we obtain a complete description of the performance of the least-squares (or “best rank one”) estimator for any rank one matrix estimation problem.
Fields of Study & Subjects
Gaussian Processes and Bayesian Inference
Bayesian Methods and Mixture Models
Statistical Mechanics and Entropy
Publication details
- Journal / source
- The Annals of Statistics
- Publisher
- Institute of Mathematical Statistics
- Volume
- 54
- Issue
- 4
- Pages
- 2031–2054
- Publication date
- 1 August 2026
- Language
- EN