Extend the theory to anisotropic and non-Gaussian data

Extend the theoretical analysis of conditional diffusion models with random-feature score networks from isotropic Gaussian data and two-eigenvalue residual covariances to anisotropic marginal distributions and non-Gaussian data.

Background

The paper analyzes memorization and generalization in conditional diffusion models using a Gaussian data model with an isotropic marginal distribution and a residual covariance having two distinct eigenvalues. The proof exploits this covariance structure by evaluating traces separately on the observed and unobserved eigenspaces.

The authors indicate that the analysis may extend to residual covariances with finitely many distinct eigenvalues, but explicitly identify extension to anisotropic marginals and non-Gaussian data as unresolved. Such an extension would broaden the theoretical results beyond the tractable Gaussian setting used for the proportional-limit calculations.