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：本文提出了一种新的基于模拟的推断方法，用于在仅能获取独立同分布样本的条件下对多维概率分布进行建模与采样，规避了传统的显式密度函数建模或马尔可夫链蒙特卡洛方法。受度量测度空间之间距离与同构性开创性工作的启发，我们提出了一种名为“可逆Gromov-Monge（RGM）”距离的新概念，并研究了如何利用RGM设计新型变换采样器以执行基于模拟的推断。我们的RGM采样器还能根据经验数据集估计两个异构度量测度空间 $(\cX, \mu, c_{\cX})$ 和 $(\cY, \nu, c_{\cY})$ 之间的最优对齐，得到的映射可近似实现一个测度 $\mu$ 向另一个测度 $\nu$ 的正向推演及其逆推演。我们研究了RGM距离的解析性质，并证实在温和条件下RGM与经典Gromov-Wasserstein距离相等。有趣的是，通过联系Brenier极分解，我们发现当合理选择 $c_{\cX}$ 和 $c_{\cY}$ 时，RGM采样器会诱导出偏向强同构性的偏差。我们还研究了该采样器的统计收敛速率、表示及优化问题，并通过合成与真实案例展示了RGM采样器的有效性。