# Concentration of the matrix-valued minimum mean-square error in optimal Bayesian inference

## TL;DR

This paper proves that the matrix-valued minimum mean-square error in Bayesian inference concentrates as the problem size grows, extending techniques from spin glass physics to various inference models.

## Contribution

It introduces a concentration result for the MMSE in vector-valued Bayesian inference, applicable to models like spiked matrices, tensors, and neural networks, under known model parameters.

## Key findings

- MMSE concentrates in large-scale Bayesian inference models
- Applicable to spiked matrix and tensor models, neural networks, and generalized linear models
- Provides theoretical foundation for mutual information formulas in these settings

## Abstract

We consider Bayesian inference of signals with vector-valued entries. Extending concentration techniques from the mathematical physics of spin glasses, we show that the matrix-valued minimum mean-square error concentrates when the size of the problem increases. Such results are often crucial for proving single-letter formulas for the mutual information when they exist. Our proof is valid in the optimal Bayesian inference setting, meaning that it relies on the assumption that the model and all its hyper-parameters are known. Examples of inference and learning problems covered by our results are spiked matrix and tensor models, the committee machine neural network with few hidden neurons in the teacher-student scenario, or multi-layers generalized linear models.