Data-Driven Priors in the Maximum Entropy on the Mean Method for Linear Inverse Problems

TL;DR

This paper develops a theoretical framework for the maximum entropy on the mean (MEM) method using data-driven priors in linear inverse problems, providing convergence guarantees and practical illustrations on image denoising tasks.

Contribution

It introduces a rigorous theoretical foundation for MEM with empirical priors, including convergence proofs and error estimates, enhancing its applicability to real-world inverse problems.

Findings

Proves almost sure convergence of empirical means in MEM.

Provides error bounds based on log-moment generating functions.

Demonstrates effectiveness on MNIST and Fashion-MNIST denoising tasks.

Abstract

We establish the theoretical framework for implementing the maximumn entropy on the mean (MEM) method for linear inverse problems in the setting of approximate (data-driven) priors. We prove a.s. convergence for empirical means and further develop general estimates for the difference between the MEM solutions with different priors $\mu$ and $\nu$ based upon the epigraphical distance between their respective log-moment generating functions. These estimates allow us to establish a rate of convergence in expectation for empirical means. We illustrate our results with denoising on MNIST and Fashion-MNIST data sets.