sbgm 0.0.8

Score-based Diffusion models in JAX.

sbgm

Score-based Diffusion models in JAX

Implementation and extension of Score-Based Generative Modeling through Stochastic Differential Equations (Song++20) and Maximum Likelihood Training of Score-Based Diffusion Models (Song++21) in jax and equinox.

[!WARNING] :building_construction: Note this repository is under construction, expect changes. :building_construction:

Score-based diffusion models

Diffusion models are deep hierarchical models for data that use neural networks to model the reverse of a diffusion process that adds a sequence of noise perturbations to the data.

A diagram (see citations) showing how to map data to a noise distribution (the prior) with an SDE, and reverse this SDE for generative modeling. One can also reverse the associated probability flow ODE, which yields a deterministic process that samples from the same distribution as the SDE. Both the reverse-time SDE and probability flow ODE can be obtained by estimating the score.

Modern cutting-edge diffusion models (see citations) express both the forward and reverse diffusion processes as a Stochastic Differential Equation (SDE).

For any SDE of the form

$$ \text{d}\boldsymbol{x} = f(\boldsymbol{x}, t)\text{d}t + g(t)\text{d}\boldsymbol{w} $$

there exists an associated ordinary differential equation (ODE)

$$ \text{d}\boldsymbol{x} = [f(\boldsymbol{x}, t)\text{d}t - \frac{1}{2}g(t)^2\nabla_{\boldsymbol{x}}\log p_t(\boldsymbol{x})]\text{d}t $$

where the trajectories of the SDE and ODE have the same marginal PDFs $p_t(\boldsymbol{x})$.

Computing log-likelihoods with diffusion models

For each SDE there exists a deterministic ODE with marginal likelihoods $p_t(\boldsymbol{x})$ that match the SDE for all time $t$

$$ \text{d}\boldsymbol{x} = [f(\boldsymbol{x}, t)\text{d}t - \frac{1}{2}g(t)^2\nabla_{\boldsymbol{x}}\log p_t(\boldsymbol{x})]\text{d}t = F(\boldsymbol{x}(t), t) $$

The continuous normalizing flow formalism allows the ODE to be expressed as

$$ \frac{\partial}{\partial t} \log p(\boldsymbol{x}(t)) = -\text{Tr}\bigg [ \frac{\partial}{\partial \boldsymbol{x}(t)} F(\boldsymbol{x}(t), t) \bigg ] $$

but note that maximum-likelihood training is prohibitively expensive for SDE based diffusion models.

Usage

Install via

pip install sbgm

to run

python main.py

Features

• Parallelised exact and approximate log-likelihood calculations,
• UNet and transformer score network implementations,
• VP, SubVP and VE SDEs (neural network $\beta(t)$ and $\sigma(t)$ functions are on the list!),
• Multi-modal conditioning (basically just optional parameter and image conditioning methods),
• Multi-device training and sampling.

Samples

[!NOTE] I haven't optimised any training/architecture hyperparameters or trained long enough here, you could do a lot better.

Flowers

Euler-Marayama sampling

ODE sampling

CIFAR10

Euler-Marayama sampling

ODE sampling

SDEs

Citations

@misc{song2021scorebasedgenerativemodelingstochastic,
      title={Score-Based Generative Modeling through Stochastic Differential Equations}, 
      author={Yang Song and Jascha Sohl-Dickstein and Diederik P. Kingma and Abhishek Kumar and Stefano Ermon and Ben Poole},
      year={2021},
      eprint={2011.13456},
      archivePrefix={arXiv},
      primaryClass={cs.LG},
      url={https://arxiv.org/abs/2011.13456}, 
}

@misc{song2021maximumlikelihoodtrainingscorebased,
      title={Maximum Likelihood Training of Score-Based Diffusion Models}, 
      author={Yang Song and Conor Durkan and Iain Murray and Stefano Ermon},
      year={2021},
      eprint={2101.09258},
      archivePrefix={arXiv},
      primaryClass={stat.ML},
      url={https://arxiv.org/abs/2101.09258}, 
}