A solvable generative model with a linear, one-step denoiser
This work provides theoretical insights into diffusion models, which is an incremental advancement for researchers in generative AI.
The authors tackled the problem of understanding diffusion models by developing an analytically tractable single-step diffusion model with a linear denoiser, deriving an explicit formula for the Kullback-Leibler divergence between generated and sampling distributions, and showing that the monotonic fall phase begins when training dataset size equals data dimension. They also explained why more diffusion steps improve quality in large-scale models based on their theoretical arguments.
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.