Eliminating the well-conditionedness assumption in diffusion-based learning of Gaussian mixtures

Establish that the diffusion-model-based score matching approach combined with piecewise polynomial approximation can efficiently learn mixtures of Gaussians without assuming the well-conditionedness constraints of bounded condition number and bounded radius of means and covariances (as specified by the parameters α, β, and R in the paper’s definition of well-conditioned mixtures), thereby matching the generality of prior moment-based methods.

Background

The main results require Gaussian mixtures to be τ-well-conditioned, meaning all component covariances have eigenvalues in [α,β] and component means/covariances lie in a ball of radius R. Although prior general-mixture works did not require such bounds, they suffered doubly exponential dependence on the number of components.

The authors ask whether their diffusion- and score-matching-based techniques can dispense with these well-conditionedness assumptions while retaining their improved runtime guarantees.