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

Abstract: We develop an analytically tractable single-step diffusion model based on a linear denoiser and present explicit formula for the Kullback-Leibler divergence between 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. Along the way, we provide a mathematically precise definition of memorization to non-memorization transition when only finite number of data points are available. It is shown that the simplified model also features this transition during the monotonic fall phase of the aforementioned Kullback-Leibler divergence. For large-scale practical diffusion models, we explain why higher number of diffusion steps enhance production quality based on the theoretical arguments presented before. In addition, we show that higher diffusion steps does not necessarily help in reducing memorization. These two facts combined suggests existence of an optimal number of diffusion steps for finite number of training samples.