This tutorial provides a beginner-friendly introduction to the concepts behind denoising diffusion models and their implementation using the JAX library. The speaker, an astrophysics researcher, breaks down the theory by first demonstrating the "forward process" of iteratively adding noise to a simple 1D Gaussian mixture until it becomes a standard normal distribution. He then introduces the "reverse process," which uses an algorithm called Langevin dynamics and a "score function" (the gradient of the log probability) to sample from the original distribution.
The tutorial progressively builds on this foundation, moving from an analytical score function to a more practical "score matching" approach where a neural network (a Multi-Layer Perceptron built with Equinox) is trained to approximate the score function of a complex 2D "Swiss roll" dataset. The key insight of modern diffusion models (DDPMs) is then introduced: making the score model time-conditioned, where it learns a different denoising map for each level of noise. This dynamic approach is shown to be far more effective at capturing the intricate details of the target distribution. Throughout the demonstration, core JAX concepts are explained, including its NumPy-like API, function transforms (vmap, grad, scan), just-in-time (JIT) compilation, and its functional programming paradigm requiring pure functions and explicit random key management.
A Beginner's Guide to Diffusion Models in JAX
0:00:28 — Introduction to Diffusion Models and JAX: The speaker introduces the two main topics of the tutorial: a conceptual overview of diffusion models, a class of generative models, and a practical guide to implementing them with JAX, a high-performance numerical computing library.
0:03:32 — The Core Concept of Diffusion: Diffusion models are based on a two-step process. The "forward process" systematically destroys data by iteratively adding noise until it becomes a simple, known distribution (e.g., Gaussian noise). The "reverse process" then learns to undo this, step-by-step, to generate new, plausible data from pure noise.
0:05:56 — What is JAX?: JAX is presented as a library that combines the familiar API of NumPy with support for hardware accelerators (GPU/TPU), automatic differentiation, and just-in-time (JIT) compilation, making it highly efficient for machine learning research.
0:19:01 — JAX Function Transforms: vmap: The tutorial introduces jax.vmap, a function transform that automatically vectorizes a function, allowing it to operate over a batch dimension of an array without needing an explicit Python loop, leading to more efficient execution.
0:21:31 — The Score Function: To reverse the diffusion process, a key piece of information is needed: the score function, which is defined as the gradient of the logarithm of the data's probability distribution. It essentially points in the direction of increasing data density.
0:22:18 — Automatic Differentiation with jax.grad: JAX's jax.grad function transform is demonstrated as a powerful tool to automatically compute the gradient of any function. This is used to derive the score function directly from the log-probability function of the toy example.
0:36:21 — Score Matching for Real-World Data: In most practical cases, the analytical form of the data distribution is unknown. "Score matching" is introduced as a method to train a model (like a neural network) to approximate the true score function by minimizing a specific loss function.
0:43:19 — The Key Idea: Time-Conditioned Models: The main innovation of modern Denoising Diffusion Probabilistic Models (DDPMs) is making the score model dependent on the diffusion time step. Instead of one static model, the system learns a different "denoising map" for each level of noise, which provides better guidance during sampling.
0:51:36 — Training the Diffusion Model: The training process involves repeatedly picking a random time step, diffusing the training data to that level of noise, and training the time-conditioned neural network to predict the noise that was added.
0:58:08 — Visualizing the Time-Dependent Score Function: A key visualization shows how the learned score function (vector field) changes over time. It starts as a simple field pointing all samples to the center and gradually morphs into a complex field that accurately captures the intricate shape of the target distribution.
1:01:08 — Final Takeaways and Comparison: The primary advantage of diffusion models is their high sample quality, while the main disadvantage is the slow, iterative nature of the sampling process. The speaker also fields questions comparing JAX's functional, JIT-compiled approach to the object-oriented style of PyTorch.