Reversible Gromov-Monge Sampler for Simulation-Based Inference

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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.