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Piecewise deterministic generative models

Andrea Bertazzi, Dario Shariatian, Umut Simsekli, Eric Moulines, Alain Durmus

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.

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 -dimensional Gaussian distribution. Promising numerical simulations support further investigations into this class of models.
Paper Structure (61 sections, 8 theorems, 87 equations, 49 figures, 5 tables, 8 algorithms)

This paper contains 61 sections, 8 theorems, 87 equations, 49 figures, 5 tables, 8 algorithms.

Table of Contents

  1. Introduction
  2. PDMP based generative models
  3. Piecewise deterministic Markov processes
  4. The Zig-Zag process
  5. The Bouncy Particle sampler
  6. Randomised Hamiltonian Monte Carlo
  7. Time reversal of PDMPs
  8. Time reversal of ZZP
  9. Time reversal of BPS
  10. Approximating the characteristics of time reversals of PDMPs
  11. Approximating the jump rates of the backward ZZP via ratio matching
  12. Approximating the characteristics of BPS and RHMC
  13. Simulating the backward process
  14. Error bound in total variation distance
  15. Numerical simulations
  16. ...and 46 more sections

Key Result

Proposition 1

Consider a non-explosive PDMP $(Z_t)_{t\geqslant 0}$ with characteristics $(\Phi,\lambda,Q)$ and initial distribution $\mu_0$ on $\mathbb{R}^D$. In addition, let $\mathrm{T}_f$ be a time horizon. Suppose that $\Phi$ is locally bounded, $(t,z)\mapsto \lambda(t,z)$ is continuous in both its variables, where $\mu_0 P_t$ stands for the distribution of $Z_t$ starting from $\mu_0$.

Figures (49)

  • Figure 1: Trace plots for ZZP (left), BPS (centre), RHMC (right). In all cases $\lambda_r=1$ and $\mathrm{T}_f = 10$.
  • Figure 2: MMD $\downarrow$ for various number of backward steps, Rose dataset.
  • Figure 3: MMD $\downarrow$, runtime (ms) per method, Rose dataset.
  • Figure 6: Generation for the ZZP trained on MNIST.
  • Figure : Fractal tree
  • ...and 44 more figures

Theorems & Definitions (11)

  • Remark 1: Noise schedule
  • Proposition 1
  • Proposition 2
  • Remark 2: Variance exploding PDMPs
  • Proposition 3
  • Theorem 1
  • Proposition 4: Time reversal of ZZP
  • Proposition 5: Time reversal of BPS
  • Remark 3
  • Proposition 6: Time reversal of RHMC
  • ...and 1 more