On the Computational and Statistical Efficiency of the Empirical Maximum Entropy on the Mean Method

2026-08-27

AI Summary

本文通过改进的稳定性分析方法，提高了经验最大熵均值法（MEM）在解决数据驱动逆问题时的统计和计算效率，达到了O(n^{-1/2})的参数收敛率。

Abstract

The Maximum Entropy on the Mean (MEM) method provides a flexible computational framework for solving inverse problems by combining data fidelity with entropy-based regularization. In practice, however, the prior distribution is typically unknown but can be estimated from data, giving rise to the empirical MEM method. We establish a parametric convergence rate of $O(n^{-1/2})$ in expectation for empirical MEM, improving upon the previously established $O(n^{-1/4})$ guarantee by King-Roskamp et al. (2026). Our proof is based on a novel stability analysis of the primal and dual optimization problems under perturbations of the underlying probability measure, relying only on foundational tools from convex analysis and probability. We further show that the MEM dual problem admits a reformulation as an expected risk minimization problem, thereby placing MEM within the modern framework of stochastic optimization and enabling scalable stochastic gradient algorithms for large-scale inverse problems. Together, these results place empirical MEM as a statistically and computationally efficient methodology for data-driven inverse problems.

Problem

• Maximum Entropy on the Mean
• inverse problems
• statistical efficiency
• computational efficiency
• empirical MEM

Innovation

• Empirical Maximum Entropy on the Mean
• Convergence Rate
• Stability Analysis
• Stochastic Optimization
• Inverse Problems