Information-theoretic limits of a multiview low-rank symmetric spiked matrix model

TL;DR

This paper rigorously determines the fundamental limits of high-dimensional symmetric matrix inference problems, advancing theoretical understanding and analytical techniques for low-rank models used in principal component analysis.

Contribution

It establishes the information-theoretic limits for multiview low-rank symmetric spiked matrix models and improves the adaptive interpolation method for analyzing low-rank estimation problems.

Findings

Derived single-letter formulas for mutual information and MMSE.

Enhanced the adaptive interpolation method for low-rank models.

Provided rigorous bounds for statistical-to-computational gaps.

Abstract

We consider a generalization of an important class of high-dimensional inference problems, namely spiked symmetric matrix models, often used as probabilistic models for principal component analysis. Such paradigmatic models have recently attracted a lot of attention from a number of communities due to their phenomenological richness with statistical-to-computational gaps, while remaining tractable. We rigorously establish the information-theoretic limits through the proof of single-letter formulas for the mutual information and minimum mean-square error. On a technical side we improve the recently introduced adaptive interpolation method, so that it can be used to study low-rank models (i.e., estimation problems of "tall matrices") in full generality, an important step towards the rigorous analysis of more complicated inference and learning models.