AI 中文总结
该研究针对经验均值最大熵（MEM）方法，将其收敛速率提升至$O(n^{-1/2})$，并将其对偶问题转化为期望风险最小化问题，使其适配随机优化框架，成为数据驱动逆问题的高效方法。
AI 中文摘要
均值最大熵（MEM）方法通过将数据保真度与基于熵的正则化相结合，为求解逆问题提供了灵活的计算框架。然而在实际应用中，先验分布通常是未知的，但可从数据中估计，由此产生了经验MEM方法。我们为经验MEM建立了期望下的参数收敛速率为$O(n^{-1/2})$，改进了King-Roskamp等人（2026）之前确立的$O(n^{-1/4})$保证。我们的证明基于对原问题和对偶优化问题在基础概率测度扰动下的新颖稳定性分析，仅依赖凸分析和概率的基础工具。我们进一步证明，MEM对偶问题可重新表述为期望风险最小化问题，从而将MEM置于现代随机优化框架中，为大规模逆问题启用可扩展的随机梯度算法。综合这些结果，经验MEM成为一种用于数据驱动逆问题的统计和计算高效的方法。
英文摘要
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.
Comments39 Pages, 7 figures
