Piecewise deterministic generative models

TL;DR

This paper introduces a new class of generative models based on piecewise deterministic Markov processes, offering explicit reverse process expressions, efficient training methods, and theoretical bounds, with promising numerical results.

Contribution

It develops a novel framework for generative modeling using PDMPs, including explicit reverse process formulas and training procedures, advancing beyond traditional diffusion-based models.

Findings

Explicit formulas for reverse jump rates and kernels.

Efficient training procedures for PDMP-based models.

Total variation bounds for data distribution approximation.

Abstract

We introduce a novel class of generative models based on piecewise deterministic Markov processes (PDMPs), a family of non-diffusive stochastic processes consisting of deterministic motion and random jumps at random times. Similarly to diffusions, such Markov processes admit time reversals that turn out to be PDMPs as well. We apply this observation to three PDMPs considered in the literature: the Zig-Zag process, Bouncy Particle Sampler, and Randomised Hamiltonian Monte Carlo. For these three particular instances, we show that the jump rates and kernels of the corresponding time reversals admit explicit expressions depending on some conditional densities of the PDMP under consideration before and after a jump. Based on these results, we propose efficient training procedures to learn these characteristics and consider methods to approximately simulate the reverse process. Finally, we provide bounds in the total variation distance between the data distribution and the resulting distribution of our model in the case where the base distribution is the standard $d$-dimensional Gaussian distribution. Promising numerical simulations support further investigations into this class of models.