Mathematical analysis of singularities in the diffusion model under the
submanifold assumption
- URL: http://arxiv.org/abs/2301.07882v3
- Date: Wed, 3 May 2023 18:44:57 GMT
- Title: Mathematical analysis of singularities in the diffusion model under the
submanifold assumption
- Authors: Yubin Lu, Zhongjian Wang, Guillaume Bal
- Abstract summary: We show that the analytical mean drift function in DDPM and the score function in SGMally blow up in the final stages of the sampling process for singular data distributions.
We derive a new target function and associated loss, which remains bounded even for singular data distributions.
- Score: 0.0
- License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
- Abstract: This paper provide several mathematical analyses of the diffusion model in
machine learning. The drift term of the backwards 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 is 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
illustrate the theoretical findings with several numerical examples.
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