Mathematical analysis of singularities in the diffusion model under the submanifold assumption
This addresses a mathematical bottleneck in diffusion models for machine learning, offering a solution for handling singular data distributions, though it appears incremental as it builds on existing frameworks.
The paper tackles the problem of singularities in diffusion models for data on lower-dimensional manifolds, showing that standard drift and score functions blow up asymptotically, and proposes a new bounded target function and loss to overcome this issue.
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.