Entropy-regularized optimal transport, which has strong links to the Schrödinger bridge problem in statistical mechanics, enjoys a variety of applications from trajectory inference to generative modeling. A major driver of renewed interest in this problem is the recent development of fast matrix-scaling algorithms\textemdash known as iterative proportional fitting or the Sinkhorn algorithm\textemdash for entropic optimal transport, which have favorable complexity over traditional approaches to the unregularized problem. Here, we take a perspective on this algorithm rooted in the thermodynamic origins of Schrödinger's problem and inspired by the modern geometric theory of diffusion: is the Sinkhorn flow (viewed in continuous-time as a mirror descent by recent results) the gradient flow of entropy in a formal Riemannian geometry? We answer this question affirmatively, finding a nonlocal Wasserstein gradient structure in the dynamics of its free marginal. This offers a physical interpretation of the Sinkhorn flow as the stochastic dynamics of a particle with law evolving by the nonlocal diffusion of a chemical potential. Simultaneously, it brings a standard suite of functional inequalities characterizing Markov diffusion processes to bear upon its geometry and convergence. We prove an entropy-energy (de Bruijn) identity, a Poincaré inequality, and a Bakry-Émery-type condition under which a logarithmic Sobolev inequality (LSI) holds and implies exponential convergence of the Sinkhorn flow in entropy. We lastly discuss computational applications such as stopping heuristics and latent-space design criteria leveraging the LSI and, returning to the physical interpretation, the possibility of natural systems whose relaxation to equilibrium inherently solves entropic optimal transport or Schrödinger bridge problems.

翻译：熵正则化最优输运与统计力学中的薛定谔桥问题有着紧密联系，在轨迹推断到生成建模等多个领域均有广泛应用。近年来，对该问题重新产生兴趣的主要推动力是快速矩阵缩放算法（即迭代比例拟合或Sinkhorn算法）的发展，该算法用于求解熵正则化最优输运问题，相较于传统非正则化问题的求解方法具有更优的复杂度。本文从薛定谔问题的热力学起源出发，并受现代扩散几何理论启发，提出以下视角：Sinkhorn流（根据近期研究结果，其在连续时间下可视为镜像下降法）是否可理解为形式黎曼几何中熵的梯度流？我们对此给出了肯定回答，在其自由边缘分布的动力学中发现了非局部Wasserstein梯度结构。这为Sinkhorn流提供了物理解释：可视为粒子在化学势非局部扩散作用下演化的随机动力学过程。同时，该结构将描述马尔可夫扩散过程的标准泛函不等式工具引入其几何与收敛性分析中。我们证明了熵-能量（de Bruijn）恒等式、庞加莱不等式，以及保证对数索博列夫不等式成立并推导Sinkhorn流熵指数收敛的Bakry-Émery型条件。最后，我们讨论了基于对数索博列夫不等式的计算应用（如停止启发式算法和隐空间设计准则），并回归物理诠释，探讨了自然系统中通过弛豫过程内在求解熵正则化最优输运或薛定谔桥问题的可能性。