score-based models, diffusion models
TL;DR: Define continuous-time stochastic processes instead of discrete steps, enabling better control and flexibility. The reverse SDE is learned using score matching, and this framework can generalize to various types of noise schedules (like the variance-exploding or variance-preserving SDEs).
In [Song et al. 2019] we discussed the role of Langevin dynamics in score matching.
To recap, Langevin dynamics consists of a stochastic differential equation (SDE) that describes the evolution of a particle in a potential field.
We can use the score function, which is the gradient of the log-density of the data distribution, and apply Langevin dynamics to generate new samples from the score function.
Let's define a perturbation kernel:
Both Score matching with Langevin dynamics (SMLD) and Denoising Diffusion Probabilistic Models (DDPM) [Ho et al. 2020] add Noise
progressively to the data over discrete time steps.
Let's derive the variance exxploding and perserving continuous-time stochastic differential equations (SDEs) for the noise perturbations from the discrete expressions.
Variance Exploding (VE) is used in SMLD and it constists of progressively increasing the variance of the noise over time.
This paper leverages VE-SDE to create robust denoising trajectories and stable training.
But how is this done?
We will define two equations of the stochastic process with standard deviation
First, let's consider a Markov chain of
Variance Preserving (VP) is used in DDPM and it constists of keeping the variance of the noise constant over time.
For VE-SDE, we define
The limit when
VE-SDE | VP-SDE | |
---|---|---|
Discrete SDE | ||
Continuous SDE | ||
dx |
The reverse time SDE [Anderson 1982] is given by:
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