Diffusion Process

A diffusion process is a type of stochastic process (random process) that models how a quantity spreads over time. Mathematically, it's often described by stochastic differential equations (SDEs), such as the Wiener process or Brownian motion.

Diffusion in Statistics and Stochastic Modeling

In statistics, diffusion processes are part of continuous-time Markov processes. They are often used to model:

• Random walks (used in finance and stock price modeling),
• Brownian motion (heat or particle diffusion),
• Ito processes (used in advanced probabilistic modeling),
• Bayesian inference in time-evolving systems.

Example: Brownian Motion

If X(t) is a process such that

X(t+Δt)=X(t)+ϵ, ϵ∼N(0,σ^2Δt)

where:

• \epsilon is a random variable drawn from a normal distribution,
• The mean of the distribution is 0, which means there's no bias (the particle is equally likely to move in any direction),
• The variance of the distribution is \sigma^2 \Delta t, which means the expected magnitude of the step increases with time \Delta t and diffusion strength \sigma.

Then X(t) performs a Brownian motion. Such models are foundational to probabilistic modeling in machine learning.

Diffusion in Machine Learning

In modern machine learning, the term “diffusion” has gained popularity due to the emergence of diffusion models which is a new class of generative models that have rapidly overtaken earlier methods like GANs (Generative Adversarial Networks) and VAEs (Variational Autoencoders) in both performance and stability.

These models are inspired by thermodynamic diffusion and stochastic processes, where data (like images or audio) is gradually corrupted by noise and then recovered through a learned denoising process.

What Are Diffusion Models in ML

Diffusion models are generative models that learn to reverse a process of gradually adding noise to data (like images) and then generate new samples by reversing that noise.

Training (Forward Process):

• Add noise to data step by step until it becomes nearly pure noise.
• Mathematically modeled as a Markov chain.

Sampling (Reverse Process):

• Learn how to reverse the noise steps to recover realistic data.
• This is learned using deep neural networks.

Example: Denoising Diffusion Probabilistic Models (DDPMs)

Denoising Diffusion Probabilistic Models (DDPMs) are a type of diffusion model which learn to remove noise from an image at each step. Once trained, they can start from random noise and generate a new image step-by-step. These models have become state-of-the-art in generating images.

Learn more about DDPMs from here.

Use-Cases:

• Image synthesis (Stable Diffusion, Imagen, MidJourney)
• Text-to-image models (DALL·E 2)
• Audio synthesis (DiffWave, Noise2Music)
• Molecular structure generation (in drug discovery)

Applications of Diffusion Concepts

1. Generative AI: As seen with diffusion models in image generation and text-to-image synthesis.
2. Autonomous Systems: Particle-based simulation and diffusion filters in robotics and SLAM (Simultaneous Localization and Mapping).
3. Reinforcement Learning: Exploration can be modeled with diffusive behavior in uncertain environments.
4. Natural Language Processing (NLP): Semantic diffusion: understanding how meanings spread in a knowledge graph.
5. Recommendation Systems: Item or content diffusion in user-item graphs for better recommendations.