Piecewise deterministic generative models

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.

• The paper introduces PDMPs as a generative modeling framework that combines deterministic trajectories with stochastic jumps.
• It characterizes time-reversal properties and derives explicit backward simulation methods for processes like ZZP, BPS, and RHMC.
• Simulation techniques using splitting schemes and normalizing flows enable efficient sample generation from complex, high-dimensional data distributions.

Overview of "Piecewise Deterministic Generative Models"

The paper in question focuses on the development and analysis of a novel class of generative models centered around Piecewise Deterministic Markov Processes (PDMPs). This research explores an alternative to diffusion-based generative models, leveraging the unique characteristics of PDMPs, which integrate deterministic motion with stochastic jumps at random times. The authors provide a comprehensive theoretical framework complemented by practical methodologies for implementing these models.

Theoretical Contributions

The paper introduces PDMPs as a basis for generative modeling, examining their fundamental structure and properties. PDMPs, initially introduced in the 1980s, encompass a broad spectrum of stochastic processes, characterized by deterministic trajectories interrupted by random jumps. This differentiates them from diffusion-based processes, which rely purely on continuous stochastic evolution.

A significant theoretical advancement presented is the characterization of time-reversal properties of PDMPs. The paper demonstrates that under appropriate conditions, the time-reversal of a PDMP retains the piecewise deterministic nature, albeit with modified characteristics. This insight is pivotal for generative modeling, as it aligns with the need to simulate processes backward—from a noise distribution to the data distribution.

Application to Known PDMPs

The authors pay particular attention to three established PDMPs: the Zig-Zag Process (ZZP), the Bouncy Particle Sampler (BPS), and the Randomised Hamiltonian Monte Carlo (RHMC). For each, they derive explicit forms of time-reversed jump rates and kernels, crucial for reconstructing data points from noise through backward simulation. Notably, these processes exhibit traits rivaling those of diffusion processes, such as scalability and reduced complexity in high-dimensional spaces, making them suitable candidates for implementation in practical generative tasks.

Numerical Approaches and Learning

On the learning front, the paper explores estimation techniques for the backward PDMP characteristics, particularly focusing on the challenging task of approximating the conditional densities of velocities. For the ZZP, the authors propose a ratio-matching technique inspired by auxiliary methodologies like score matching. Normalizing flows are utilized to model conditional distributions in BPS and RHMC, leveraging their capability for both density estimation and sampling.

Simulation and Discretization

The practical implementation of these backward processes is addressed through splitting schemes that discretize the simulation of PDMPs. These schemes approximate the evolutionary dynamics of PDMPs with potential applications in efficiently generating samples from complex data distributions.

Implications and Future Work

This exploration into PDMP-based generative models offers several implications. Theoretically, it enriches the understanding of PDMPs in the context of machine learning, extending their applicability beyond traditional fields. Practically, the proposed framework could compete with or complement existing diffusion-based models, particularly in settings where the computational cost of handling high-dimensional data is significant.

The paper concludes by highlighting future research opportunities, including potential hardware optimizations and improved network architectures for better scalability. This work lays foundational insights that might inspire further refinements and applications in generative modeling, especially across domains where the characteristics of PDMPs align naturally with data generation tasks.

In summary, the paper delivers a comprehensive and technically robust exposition on piecewise deterministic processes as generative models, establishing ground for a novel direction in probabilistic modeling and synthetic data generation.