This paper introduces a new simulation-based inference procedure to model and sample from multi-dimensional probability distributions given access to i.i.d.\ samples, circumventing the usual approaches of explicitly modeling the density function or designing Markov chain Monte Carlo. Motivated by the seminal work on distance and isomorphism between metric measure spaces, we propose a new notion called the Reversible Gromov-Monge (RGM) distance and study how RGM can be used to design new transform samplers to perform simulation-based inference. Our RGM sampler can also estimate optimal alignments between two heterogeneous metric measure spaces $(\cX, \mu, c_{\cX})$ and $(\cY, \nu, c_{\cY})$ from empirical data sets, with estimated maps that approximately push forward one measure $\mu$ to the other $\nu$, and vice versa. We study the analytic properties of the RGM distance and derive that under mild conditions, RGM equals the classic Gromov-Wasserstein distance. Curiously, drawing a connection to Brenier's polar factorization, we show that the RGM sampler induces bias towards strong isomorphism with proper choices of $c_{\cX}$ and $c_{\cY}$. Statistical rate of convergence, representation, and optimization questions regarding the induced sampler are studied. Synthetic and real-world examples showcasing the effectiveness of the RGM sampler are also demonstrated.
翻译:本文引入了一种新的基于模拟的推论程序, 用于建模和采样的多维概率分布模型和样本, 提供i. d.\ 样本, 绕过明确模拟密度函数或设计 Markov 链子的通常方法, 或设计 Markov 链子 Monte Carlo。 受关于距离和度量空间之间形态化的原始工作驱动, 我们提出了一个名为Reversible Gromov- Monge(RGM) 距离的新概念, 并研究如何使用RGM设计新的变异采样器来进行基于模拟的推断。 我们的RGM 取样器还可以估计两个混杂计量空间 $(\ cX,\ mu, c ⁇ cX} 美元和 美元(cy) 链路边际抽样分析仪的典型的距离, 并展示了与精确度值的精确度值的精确度 。 我们的研究RGM相当于典型的Gromov- Wserstein 样本的距离。