On the Relation Between Linear Diffusion and Power Iteration
This provides theoretical insights into diffusion models for researchers, though it is incremental as it extends known linear results to low-rank data and non-linear cases.
The paper tackles the problem of understanding diffusion models by analyzing the linear case, showing that the generation process converges to the leading eigenvector of the data, similar to power iteration, with low frequencies emerging earlier based on eigenvalues.
Recently, diffusion models have gained popularity due to their impressive generative abilities. These models learn the implicit distribution given by the training dataset, and sample new data by transforming random noise through the reverse process, which can be thought of as gradual denoising. In this work, we examine the generation process as a ``correlation machine'', where random noise is repeatedly enhanced in correlation with the implicit given distribution. To this end, we explore the linear case, where the optimal denoiser in the MSE sense is known to be the PCA projection. This enables us to connect the theory of diffusion models to the spiked covariance model, where the dependence of the denoiser on the noise level and the amount of training data can be expressed analytically, in the rank-1 case. In a series of numerical experiments, we extend this result to general low rank data, and show that low frequencies emerge earlier in the generation process, where the denoising basis vectors are more aligned to the true data with a rate depending on their eigenvalues. This model allows us to show that the linear diffusion model converges in mean to the leading eigenvector of the underlying data, similarly to the prevalent power iteration method. Finally, we empirically demonstrate the applicability of our findings beyond the linear case, in the Jacobians of a deep, non-linear denoiser, used in general image generation tasks.