A solvable generative model with a linear, one-step denoiser

TL;DR

This paper introduces an analytically solvable single-step diffusion model with a linear denoiser, providing explicit formulas for divergence and insights into diffusion step effects on model quality.

Contribution

It presents a new tractable diffusion model with a linear denoiser and derives explicit formulas for divergence, explaining effects of diffusion time, noise scale, and dataset size.

Findings

Explicit formula for Kullback-Leibler divergence between generated and true distribution.

Monotonic divergence fall begins when dataset size equals data dimension.

More diffusion steps improve output quality in large-scale models.

Abstract

We develop an analytically tractable single-step diffusion model based on a linear denoiser and present an explicit formula for the Kullback-Leibler divergence between the generated and sampling distribution, taken to be isotropic Gaussian, showing the effect of finite diffusion time and noise scale. Our study further reveals that the monotonic fall phase of Kullback-Leibler divergence begins when the training dataset size reaches the dimension of the data points. Finally, for large-scale practical diffusion models, we explain why a higher number of diffusion steps enhances production quality based on the theoretical arguments presented before.