diffusionjax

diffusionjax is a simple, accessible introduction to diffusion models, also known as score-based generative models (SGMs). It is implemented in Python via the autodiff framework, JAX. In particular, diffusionjax uses the Flax library for the neural network approximator of the score. diffusionjax focusses on the continuous time formulation during training.

The development of diffusionjax has been supported by The Alan Turing Institute through the Theory and Methods Challenge Fortnights event "Accelerating generative models and nonconvex optimisation", which took place on 6-10 June 2022 and 5-9 Sep 2022 at The Alan Turing Institute headquarters.

Thank you to nPlan, who are supporting this project.

Contents:

• Installation
• Examples
  • Introduction to diffusion models
• Does haves
• Doesn't haves
• References

Installation

The package requires Python 3.8+. First, it is recommended to create a new python virtual environment. diffusionjax depends on JAX. Because the JAX installation is different depending on your CUDA version, diffusionjax does not list JAX as a dependency in setup.py. First, follow these instructions to install JAX with the relevant accelerator support. To run the examples, you may optionally need to install optax, orbax-checkpoint, torch[cpu] and mlkernels, which the package depends on only through the examples given. Then, pip install diffusionjax or for developers,

• Clone the repository git clone git@github.com:bb515/diffusionjax.git
• Install using pip pip install -e . from the root directory of the repository (see the setup.py for the requirements that this command installs).

Examples

Introduction to diffusion models

Run the example by typing

python examples/example.py:
  --config: Training configuration.
    (default: './configs/example.py')
  --workdir: Working directory
    (default: './examples/')

on the command line from the root directory of the repository.

• config is the path to the config file. The default config files are provided in configs/. They are formatted according to ml_collections.
• workdir is the path that stores all artifacts of one experiment, like checkpoints, samples, and evaluation results via wandb.

The example is based off the Jupyter notebook by Jakiw Pidstrigach, a tutorial on the theoretical and implementation aspects of diffusion models.

>>> num_epochs = 4000
>>> num_samples = 8
>>> samples = sample_circle(num_samples)
>>> N = samples.shape[1]
>>> plot_scatter(samples=samples, index=(0, 1), fname="samples", lims=((-3, 3), (-3, 3)))
>>> rng = random.PRNGKey(2023)

>>> # Get variance preserving (VP) a.k.a. time-changed Ohrnstein Uhlenbeck (OU) sde model
>>> sde = VP()
>>>
>>> def log_hat_pt(x, t):
>>>     """
>>>     Empirical distribution score.
>>>
>>>     Args:
>>>     x: One location in $\mathbb{R}^2$
>>>     t: time
>>>     Returns:
>>>     The empirical log density, as described in the Jupyter notebook
>>>     .. math::
>>>         \hat{p}_{t}(x)
>>>     """
>>>     mean, std = sde.marginal_prob(samples, t)
>>>     potentials = jnp.sum(-(x - mean)**2 / (2 * std**2), axis=1)
>>>     return logsumexp(potentials, axis=0, b=1/num_samples)
>>>
>>> # Get a jax grad function, which can be batched with vmap
>>> nabla_log_hat_pt = jit(vmap(grad(log_hat_pt), in_axes=(0, 0), out_axes=(0)))
>>>
>>> # Running the reverse SDE with the empirical drift
>>> plot_score(score=nabla_log_hat_pt, t=0.01, area_min=-3, area_max=3, fname="empirical score")

>>> sampler = get_sampler((5760, N), EulerMaruyama(sde.reverse(nabla_log_hat_pt)))
>>> rng, *sample_rng = random.split(rng, 2)
>>> q_samples = sampler(jnp.array(sample_rng))
>>> q_samples = q_samples.reshape(5760, N)
>>> plot_heatmap(samples=q_samples, area_min=-3, area_max=3, fname="heatmap empirical score")

>>> # What happens when I perturb the score with a constant?
>>> perturbed_score = lambda x, t: nabla_log_hat_pt(x, t) + 1
>>> sampler = get_sampler((5760, N), EulerMaruyama(sde.reverse(perturbed_score)))
>>> rng, *sample_rng = random.split(rng, 2)
>>> q_samples = sampler(jnp.array(sample_rng))
>>> q_samples = q_samples.reshape(5760, N)
>>> plot_heatmap(samples=q_samples, area_min=-3, area_max=3, fname="heatmap bounded perturbation")

>>> # Neural network training via score matching
>>> batch_size=16
>>> score_model = MLP()
>>> # Initialize parameters
>>> params = score_model.init(step_rng, jnp.zeros((batch_size, N)), jnp.ones((batch_size,)))
>>> # Initialize optimizer
>>> opt_state = optimizer.init(params)
>>> # Get loss function
>>> solver = EulerMaruyama(sde)
>>> loss = get_loss(
>>>     sde, solver, score_model, score_scaling=True, likelihood_weighting=False)
>>> # Train with score matching
>>> score_model, params, opt_state, mean_losses = retrain_nn(
>>>     update_step=update_step,
>>>     num_epochs=num_epochs,
>>>     step_rng=step_rng,
>>>     samples=samples,
>>>     score_model=score_model,
>>>     params=params,
>>>     opt_state=opt_state,
>>>     loss=loss,
>>>     batch_size=batch_size)
>>> # Get trained score
>>> trained_score = get_score(sde, score_model, params, score_scaling=True)
>>> plot_score(score=trained_score, t=0.01, area_min=-3, area_max=3, fname="trained score")

>>> solver = EulerMaruyama(sde.reverse(trained_score))
>>> sampler = get_sampler((720, N), solver, stack_samples=False)
>>> rng, *sample_rng = random.split(rng, 2)
>>> q_samples = sampler(jnp.array(sample_rng))
>>> q_samples = q_samples.reshape(720, N)
>>> plot_heatmap(samples=q_samples, area_min=-3, area_max=3, fname="heatmap trained score")

>>>  # Condition on one of the coordinates
>>>  y = jnp.array([-0.5, 0.0])
>>>  mask = jnp.array([1., 0.])
>>>  # Get inpainter
>>>  sampler = get_sampler(sampling_shape,
                           solver,
                           Inpainted(sde.reverse(trained_score), mask, y),
                           inverse_scaler=inverse_scaler,
                           stack_samples=False,
                           denoise=True)
>>>  q_samples, _ = sampler(sample_rng)
>>>  q_samples = q_samples.reshape(sampling_shape)
>>>  plot_heatmap(samples=q_samples, area_bounds=[-3., 3.], fname="heatmap inpainted")

Does haves

• Training scores on (possibly, image) data and sampling from the generative model. Also inverse problems, such as inpainting.
• jit multiple training steps together to improve training speed at the cost of more memory usage. This can be set via config.training.n_jitted_steps.
• Not many lines of code.
• Bayesian inversion (inverse problems) with linear observation maps.
• Easy to use, extendable. Get started with the example, provided.
• Implements a JAX port of the model and loss from Analyzing and Improving the Training Dynamics of Diffusion Models

Doesn't haves

• Geometry other than Euclidean space, such as Riemannian manifolds.
• Diffusion in a latent space.
• Augmented with critically-damped Langevin diffusion.

References

This is the implementation for the paper Tweedie Moment Projected Diffusions for Inverse Problems by Benjamin Boys, Mark Girolami, Jakiw Pidstrigach, Sebastian Reich, Alan Mosca and O. Deniz Akyildiz.