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

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.