标题： 连续时间Sinkhorn动力学中的收缩和熵产生 显示英文标题

标题： Contraction and entropy production in continuous-time Sinkhorn dynamics

摘要： 最近，有限正则化参数$\varepsilon$下Sinkhorn算法的消失步长极限被证明是在概率测度空间中的镜像下降。 我们给出了由镜像Hessian诱导的两个时间依赖度量中的$L^2$收缩准则，这些准则退化为某些条件期望算子的强制性。 然后，我们给出了Sinkhorn流的熵产生率的精确表达式，该表达式之前仅知是非正的。 检查此速率表明，扩散过程的标准半群分析系统地扩展到了Sinkhorn流。 我们证明了该流在目标边缘上诱导了一个可逆的马尔可夫动力学，作为Onsager梯度流。 我们定义了与其（非局部）无穷小生成器相关的狄利克雷形式，证明了其Poincaré不等式，并且当$\varepsilon > 0$时，证明了沿Sinkhorn流的谱间隙严格为正。 最后，我们证明了熵衰减是指数级的当且仅当满足对数Sobolev不等式（LSI）。 为了说明，我们给出了Sinkhorn LSI的两个直接实际用例：作为生成模型训练潜在空间的设计原则，以及作为离散时间算法的停止启发式。 显示英文摘要

摘要： Recently, the vanishing-step-size limit of the Sinkhorn algorithm at finite regularization parameter $\varepsilon$ was shown to be a mirror descent in the space of probability measures. We give $L^2$ contraction criteria in two time-dependent metrics induced by the mirror Hessian, which reduce to the coercivity of certain conditional expectation operators. We then give an exact identity for the entropy production rate of the Sinkhorn flow, which was previously known only to be nonpositive. Examining this rate shows that the standard semigroup analysis of diffusion processes extends systematically to the Sinkhorn flow. We show that the flow induces a reversible Markov dynamics on the target marginal as an Onsager gradient flow. We define the Dirichlet form associated to its (nonlocal) infinitesimal generator, prove a Poincar\'e inequality for it, and show that the spectral gap is strictly positive along the Sinkhorn flow whenever $\varepsilon > 0$. Lastly, we show that the entropy decay is exponential if and only if a logarithmic Sobolev inequality (LSI) holds. We give for illustration two immediate practical use-cases for the Sinkhorn LSI: as a design principle for the latent space in which generative models are trained, and as a stopping heuristic for discrete-time algorithms.