Forward-Evolution Error Analysis and Adaptive Design for Matrix-Valued Diffusion Models

Abstract

Diffusion models learn to reverse a predefined corruption process, but sampling still requires a costly time discretization and depends on the chosen noise schedule. We study these two issues for variance-preserving diffusions with matrix-valued schedules. Our analysis transfers reverse-time discretization errors to the forward corruption law and treats two numerical schemes within a common framework. The first freezes the score and yields, through a matrix-sensitive local comparison and forward information dissipation, an ambient-dimensional step complexity with leading factor $d/{\epsilon }^2$ for KL accuracy ${\epsilon }^2$. The second keeps the known Gaussian drift exact and freezes the posterior mean. For data of metric-entropy dimension $k$, a forward Markov identity, an anisotropic covering estimate, and Stieltjes integration by parts give the corresponding factor $k\log k/{\epsilon }^2$. In both cases, the proof identifies a local error, accumulates it through the forward evolution, and inserts the result into a common KL decomposition. The local errors further provide directional criteria for matrix schedules and an asymptotically optimal square-root adaptive grid. A high-dimensional Gaussian-mixture experiment illustrates the resulting schedule and grid improvements.