提出更高效的统计与计算方法，解决数据驱动的逆问题。
On the Computational and Statistical Efficiency of the Empirical Maximum Entropy on the Mean Method
- 基于凸分析和概率论，建立新稳定性分析框架。
- 实证最大熵方法收敛速度达O(n⁻¹/²)，优于此前O(n⁻¹⁴)。
- 将对偶问题重构成期望风险最小化，支持大规模随机优化。
最大熵均值（MEM）方法通过结合数据保真度与熵正则化，为求解逆问题提供灵活的计算框架。实践中，先验分布通常未知，可从数据中估计，从而引出实证MEM方法。本文建立了实证MEM在期望下的参数收敛率为O(n⁻¹/²)，优于先前由King-Roskamp等人（2026）给出的O(n⁻¹⁴)。证明基于对原始与对偶优化问题在概率测度扰动下的新颖稳定性分析，仅依赖凸分析与概率论基础工具。进一步表明，MEM对偶问题可重构为期望风险最小化问题，使MEM纳入现代随机优化框架，支持大规模逆问题的可扩展随机梯度算法。这些结果共同确立了实证MEM在数据驱动逆问题中的统计与计算高效性。
原文摘要 · Abstract (English)
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.
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