  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.
