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Piecewise deterministic generative models
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Introduces piecewise deterministic Markov process (PDMP) generative models using Zig-Zag, Bouncy Particle Sampler, and Randomised Hamiltonian Monte Carlo as noising processes, with time reversals derived explicitly and learned via ratio matching or normalising flows.
Problem
Diffusion-based generative models rely on diffusion noising processes; the paper asks whether non-diffusive PDMPs can serve as forward noising processes whose time reversals yield a generative sampler.
Approach
Characterise the time reversal of a PDMP as another PDMP with reversed deterministic motion and jump rates/kernels given by ratios of conditional velocity densities; learn these ratios (ratio matching for ZZP, normalising flows for BPS/RHMC) and simulate the backward process with splitting schemes.
Key results
- ZZP achieves lowest MMD on Checkerboard (0.81e-3), Fractal tree (1.12e-3), Olympic rings (1.43e-3) and Rose (0.90e-3), versus 2.49e-3, 8.04e-3, 2.03e-3, 6.77e-3 for i-DDPM.
- On Gaussian grid, RHMC achieves the lowest MMD (4.01e-3) versus 23.19e-3 for i-DDPM.
- Total variation bound of the form Ce^{-gamma T_f} + 4 M T_f d under bounded expected rate-estimation error for ZZP.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use ZZP instead of i-DDPM | 2D synthetic generation · Checkerboard | MMD: 2.49 ± 0.98 -> 0.81 ± 0.19 | Helped |
| Use ZZP instead of i-DDPM | 2D synthetic generation · Fractal tree | MMD: 8.04 ± 5.58 -> 1.12 ± 0.58 | Helped |
| Use RHMC instead of i-DDPM | 2D synthetic generation · Gaussian grid | MMD: 23.19 ± 9.72 -> 4.01 ± 3.32 | Helped |
| Use ZZP instead of i-DDPM | 2D synthetic generation · Olympic rings | MMD: 2.03 ± 1.60 -> 1.43 ± 0.86 | Helped |
| Use ZZP instead of i-DDPM | 2D synthetic generation · Rose | MMD: 6.77 ± 5.81 -> 0.90 ± 0.35 | Helped |
| Use PDMP-based generative models instead of i-DDPM | 2D synthetic generation · Five toy distributions | MMD | Mixed |
Surprising findings
- ZZP does not perform as well as BPS and RHMC for a small number of reverse steps, since its velocities are constrained to {-1,1}, though it is best given enough steps.
Limitations
- Architecture sensitivity; high-dimensional learning of ZZP ratios is costly (cost linear in d)
- Experiments limited to simple 2D synthetic datasets with only preliminary MNIST results
- Discretisation error of splitting schemes neglected in the TV bound
Experimental setup
Datasets: Checkerboard Fractal tree Gaussian grid Olympic rings Rose MNIST
Benchmarks: Checkerboard Fractal tree Gaussian grid Olympic rings Rose
Metrics: kernel maximum mean discrepancy
Baselines: i-DDPM
Builds on: denoising diffusion probabilistic models time reversal of Markov jump processes Zig-Zag process Bouncy Particle Sampler Randomised Hamiltonian Monte Carlo
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