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Reversible Gromov-Monge Sampler for Simulation-Based Inference

YoonHaeng Hur, Wenxuan Guo, Tengyuan Liang

TL;DR

This work introduces the Reversible Gromov-Monge ($RGM$) distance to enable simulation-based inference between heterogeneous metric measure spaces. By decoupling transport maps into a forward map $F$ and a backward map $B$ with a binding constraint, the authors design a transform-sampling procedure that avoids explicit density modeling and MCMC. They establish theoretical properties, including the metric nature of $RGM$, conditions under which it matches Gromov-Wasserstein, and non-asymptotic convergence rates via RKHS-based representations and pseudo-dimensions. The framework is instantiated with two optimization paradigms (non-convex gradient descent and infinite-dimensional convex representer form) and demonstrated on Gaussian data and MNIST-like digits, illustrating effective approximate isomorphisms and practical likelihood-free inference capabilities. Overall, the approach offers a principled, scalable path to sampling and alignment across heterogeneous spaces with a clear inductive bias toward strong isomorphisms, guided by Brenier's polar factorization insights.

Abstract

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 $(\cX, μ, c_{\cX})$ and $(\cY, ν, c_{\cY})$ from empirical data sets, with estimated maps that approximately push forward one measure $μ$ to the other $ν$, 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.

Reversible Gromov-Monge Sampler for Simulation-Based Inference

TL;DR

This work introduces the Reversible Gromov-Monge () distance to enable simulation-based inference between heterogeneous metric measure spaces. By decoupling transport maps into a forward map and a backward map with a binding constraint, the authors design a transform-sampling procedure that avoids explicit density modeling and MCMC. They establish theoretical properties, including the metric nature of , conditions under which it matches Gromov-Wasserstein, and non-asymptotic convergence rates via RKHS-based representations and pseudo-dimensions. The framework is instantiated with two optimization paradigms (non-convex gradient descent and infinite-dimensional convex representer form) and demonstrated on Gaussian data and MNIST-like digits, illustrating effective approximate isomorphisms and practical likelihood-free inference capabilities. Overall, the approach offers a principled, scalable path to sampling and alignment across heterogeneous spaces with a clear inductive bias toward strong isomorphisms, guided by Brenier's polar factorization insights.

Abstract

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 and from empirical data sets, with estimated maps that approximately push forward one measure to the other , 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 and . 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.

Paper Structure

This paper contains 44 sections, 24 theorems, 229 equations, 9 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Main contributions
  3. Organization
  4. Related Literature
  5. Background
  6. A Brief Overview of Optimal Transport Theory
  7. Gromov-Wasserstein and Gromov-Monge Distances
  8. Summary of Results
  9. Metric Properties of Reversible Gromov-Monge
  10. Transform Sampling via RGM
  11. Statistical Rate of Convergence
  12. Overview of assumptions
  13. Convex Formulation and Representer Theorem
  14. Experiments
  15. Gaussian distributions
  16. ...and 29 more sections

Key Result

Theorem 1

Let $\mathcal{M}$ be the collection of all network spaces $(\mathcal{X}, \mu, c_{\mathcal{X}})$ such that $c_{\mathcal{X}} = d_{\mathcal{X}}$. Also, let $\mathcal{M}/_{\cong}$ be the collection of all equivalence classes of $\mathcal{M}$ induced by $\cong$. Then, GW satisfies the three metric axioms

Figures (9)

  • Figure 1: Gaussian experiment: $m = n = 1000$ and $\lambda_1 = \lambda_2 = \lambda_3 = 1$. (a) shows $\{\tilde{y}_j\}_{j = 1}^{400}$ versus $\{\widehat{F}(\tilde{x}_i)\}_{i = 1}^{400}$, where $\{\tilde{y}_j\}_{j = 1}^{400}$ and $\{\tilde{x}_i\}_{i = 1}^{400}$ are i.i.d. from $\nu = N(0, \Sigma)$ and $\mu = N(0, I_2)$, respectively; they are new samples independent from $\{y_j\}_{j = 1}^{1000}$ and $\{x_i\}_{i = 1}^{1000}$ used in \ref{['eqn:1']}. (b) shows the points $\{(c_{\mathcal{X}}(\tilde{x}_i, \tilde{x}_{i'}), c_{\mathcal{Y}}(\widehat{F}(\tilde{x}_i), \widehat{F}(\tilde{x}_{i'})))\}_{i, i' = 1}^{40}$ and a straight line $y = x$.
  • Figure 2: (a) and (b) are generated by transforming new i.i.d. samples from $\mu$ using $\widehat{F}$: (a) from $\mu = N(0, I_2)$ with MMDs and (b) from $\mu = N(0, I_4)$ with Sinkhorn divergences. (c) shows real MNIST images.
  • Figure 3: (a) is generated by applying $\widehat{B}$ to $500$ out-of-sample MNIST images, i.i.d. $\{\tilde{y}_j\}_{j = 1}^{500}$ from $\nu$. (b) shows the points $\{(c_\mathcal{X}(\widehat{B}(\tilde{y}_{j}), \widehat{B}(\tilde{y}_{j'})), c_\mathcal{Y}(\tilde{y}_j, \tilde{y}_{j'}))\}_{j, j' = 1}^{50}$ and a straight line $y = x$.
  • Figure 4: Optimal transport maps by Brenier's theorem
  • Figure 5: Generalization of Figure \ref{['fig:diagrams_cost']}(b) via Brenier's polar factorization.
  • ...and 4 more figures

Theorems & Definitions (54)

  • Definition 1
  • Definition 2
  • Remark
  • Definition 3
  • Theorem 1: Lemma 1.10 of sturm_2012
  • Definition 4
  • Definition 5
  • Remark
  • Theorem 2
  • Remark
  • ...and 44 more