Latent Process Generator Matching

TL;DR

This paper introduces a general framework called latent process generator matching, which extends existing generator matching theory to time-dependent latent processes, enabling better modeling of complex stochastic dynamics.

Contribution

It generalizes generator matching to include time-dependent latent processes, broadening the applicability of the theory beyond static latent variables.

Findings

The framework allows learning generators with the same marginals as projected processes.

It subsumes previous results on discrete latent processes.

Extends generator matching to richer, time-dependent latent dynamics.

Abstract

Many recent flow-matching and diffusion-style generative models rely on auxiliary stochastic dynamics during training: a richer process is simulated to define conditional targets, but the auxiliary state is either intractable to sample at generation time or simply not part of the desired output. Existing Generator Matching theory formalises conditioning on static latent random variables, and several recent papers prove special cases of projection results for particular augmented-state constructions. We introduce latent process generator matching, a general framework that treats the observed generative state as a deterministic image $X_t=\Phi(Y_t)$ of a tractable Markov process $Y_t$. We show that in this setting one may learn the generator of a stochastic process on the image space which has the same one-time marginal distributions as the projected process. This generalizes and subsumes the discrete latent process results from the literature, and extends Generator Matching from static latent variables to a rich family of time-dependent latent conditional processes.