Deterministic Dynamics of Sampling Processes in Score-Based Diffusion Models with Multiplicative Noise Conditioning

TL;DR

This paper provides a theoretical analysis of score-based diffusion models with multiplicative noise conditioning, explaining why they perform well despite limited representational capacity.

Contribution

It offers a deterministic differential equation perspective to understand the dynamics of these models, revealing insights into their practical effectiveness.

Findings

The models perform well despite limited expressiveness.

Differential equations explain the models' success.

Theoretical insights align with empirical observations.

Abstract

Score-based diffusion models generate new samples by learning the score function associated with a diffusion process. While the effectiveness of these models can be theoretically explained using differential equations related to the sampling process, previous work by Song and Ermon (2020) demonstrated that neural networks using multiplicative noise conditioning can still generate satisfactory samples. In this setup, the model is expressed as the product of two functions: one depending on the spatial variable and the other on the noise magnitude. This structure limits the model's ability to represent a more general relationship between the spatial variable and the noise, indicating that it cannot fully learn the correct score. Despite this limitation, the models perform well in practice. In this work, we provide a theoretical explanation for this phenomenon by studying the deterministic dynamics of the associated differential equations, offering insight into how the model operates.