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.

This repository provides a lightweight library of models, sampling and likelihood routines. Suitable for likelihood-free or emulation based approaches. Tested and typed code to ensure reliable and benchmarkable training and inference.

Warning

🏗️ Note this repository is under construction, expect changes. 🏗️

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.

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

---

A diagram 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.

---

For any SDE of the form

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

the reverse of the SDE from noise to data is given by

$$ \text{d}x = [f(x, t) - g(t)^2\nabla_{x}\log p_t(x)]\text{d}t + g(t)\text{d}w. $$

For every SDE there exists an associated ordinary differential equation (ODE)

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

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

The Stein score of the marginal probability distributions over $t$ is approximated with a neural network $\nabla_{x}\log p_t(x)\approx s_{\theta}(x(t), t)$. The parameters of the neural network are fit by minimising the score-matching loss.

Computing log-likelihoods with diffusion models

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

$$ \text{d}x = [f(x, t)\text{d}t - \frac{1}{2}g(t)^2\nabla_{x}\log p_t(x)]\text{d}t = f'(x(t), t)\text{d}t. $$

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

$$ \frac{\partial}{\partial t} \log p(x(t)) = \nabla_{x} \cdot f'(x(t), t), $$

which gives the log-likelihood of a datapoint $x$ as

$$ \log p(x(0)) = \log p(x(T)) + \int_{t=0}^{t=T}\text{d}t ; \nabla_{x}\cdot f'(x, t). $$

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

Usage

Install via

pip install sbgm

and for the examples, run

pip install .[examples] 

To fit a diffusion model to the cifar10 image dataset, try something like

import sbgm
import configs

datasets_path = "./"
root_dir = "./"

config = configs.cifar10_config()

key = jr.key(config.seed)
data_key, model_key, train_key = jr.split(key, 3)

dataset = sbgm.data.cifar10(datasets_path, data_key)

sharding = sbgm.shard.get_sharding()
    
# Diffusion model 
model = sbgm.models.get_model(
    model_key, 
    config.model.model_type, 
    dataset.data_shape, 
    dataset.context_shape, 
    dataset.parameter_dim,
    config
)

# Stochastic differential equation (SDE)
sde = sbgm.sde.get_sde(config.sde)

# Fit model to dataset
model = sbgm.train.train(
    train_key,
    model,
    sde,
    dataset,
    config,
    sharding=sharding,
    save_dir=root_dir
)

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),
• Checkpointing for optimiser and model,
• 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}, 
}