首次证明迭代马尔可夫拟合在有限时间内指数收敛，为生成模型提供理论保障。
Exponential Convergence Guarantees for Iterative Markovian Fitting
- 提出新收缩分析方法，突破传统渐近收敛限制。
- 在对数凹与弱对数凹分布下，实现指数级收敛速度。
- 为扩散型薛定谔桥模型提供理论基础，适合关注生成模型的学者。
薛定谔桥问题已成为计算最优传输和生成建模中的核心工具。针对该问题，理想方法如迭代比例调整和迭代马尔可夫拟合（IMF）已被提出，同时出现了诸如扩散薛定谔桥及其匹配变体（DSBM）等实用近似方法。尽管先前研究已建立IMF的渐近收敛性，但缺乏定量的非渐近收敛保证。本文在参考测度和边缘分布满足较弱结构假设的前提下，首次建立了IMF在足够长时域下的非渐近指数收敛性。结果涵盖两类关键情形：边缘分布为对数凹或弱对数凹的情况。分析依赖于对马尔可夫投影算子的新收缩结果，为DSBM提供了理论支撑。
原文摘要 · Abstract (English)
The Schrödinger Bridge (SB) problem has become a fundamental tool in computational optimal transport and generative modeling. To address this problem, ideal methods such as Iterative Proportional Fitting and Iterative Markovian Fitting (IMF) have been proposed-alongside practical approximations like Diffusion Schrödinger Bridge and its Matching (DSBM) variant. While previous work have established asymptotic convergence guarantees for IMF, a quantitative, non-asymptotic understanding remains unknown. In this paper, we provide the first non-asymptotic exponential convergence guarantees for IMF under mild structural assumptions on the reference measure and marginal distributions, assuming a sufficiently large time horizon. Our results encompass two key regimes: one where the marginals are log-concave, and another where they are weakly log-concave. The analysis relies on new contraction results for the Markovian projection operator and paves the way to theoretical guarantees for DSBM.
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