Mathematical analysis of singularities in the diffusion model under the submanifold assumption

TL;DR

This paper provides a mathematical analysis of diffusion models in machine learning, revealing the blow-up of drift and score functions on singular data distributions and proposing a bounded alternative target function.

Contribution

It introduces a new bounded target function and loss for diffusion models that remain stable on singular data distributions, addressing a key challenge in the field.

Findings

Score and drift functions blow up on lower-dimensional manifolds

Proposed new target function remains bounded for singular distributions

Numerical examples validate theoretical predictions

Abstract

This paper concerns the mathematical analyses of the diffusion model in machine learning. The drift term of the backward sampling process is represented as a conditional expectation involving the data distribution and the forward diffusion. The training process aims to find such a drift function by minimizing the mean-squared residue related to the conditional expectation. Using small-time approximations of the Green's function of the forward diffusion, we show that the analytical mean drift function in DDPM and the score function in SGM asymptotically blow up in the final stages of the sampling process for singular data distributions such as those concentrated on lower-dimensional manifolds, and are therefore difficult to approximate by a network. To overcome this difficulty, we derive a new target function and associated loss, which remains bounded even for singular data distributions. We validate the theoretical findings with several numerical examples.