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

arXiv: https://arxiv.org/abs/2608.27705 · HTML: https://mlsanity.com/paper/2608.27705/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 4 of its 38 references that are in the map.
- First posted on arXiv 27 Aug 2026.

## Builds on (most cited first, 4 of 4)

- [Stability and Convergence of Stochastic Gradient Clipping: Beyond Lipschitz Continuity and Smoothness](https://mlsanity.com/paper/2102.06489/) (2021)
- [A study of convex convex-composite functions via infimal convolution with applications](https://mlsanity.com/paper/1907.08318/) (2019)
- [Maximum Entropy on the Mean and the Cram'er Rate Function in Statistical Estimation and Inverse Problems: Properties, Models, and Algorithms](https://mlsanity.com/paper/2211.05205/) (2022)
- [Data-Driven Priors in the Maximum Entropy on the Mean Method for Linear Inverse Problems](https://mlsanity.com/paper/2412.17916/) (2024)

## Abstract (arXiv)

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.