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

2026-08-15

AI Summary

This study addresses the high temporal discretization costs and noise schedule dependencies in matrix-valued diffusion model sampling by shifting reverse discretization errors to the forward process under a unified numerical analysis framework. By establishing a forward evolution error accumulation framework and leveraging Markov identities alongside KL divergence decomposition, we derive step-size complexity bounds with respect to ambient and metric entropy dimensions. Accordingly, we propose asymptotically optimal adaptive meshing and matrix scheduling criteria, theoretically deriving the optimal step-size factor. Experiments on Gaussian mixtures validate that this approach significantly reduces sampling complexity while enhancing schedule effectiveness, offering a principled solution for efficient generation in 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 $d/\varepsilon^2$ for KL accuracy $\varepsilon^2$. 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 $k\log k/\varepsilon^2$. 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.

Problem

• Matrix-Valued Diffusion Models
• Time Discretization
• Noise Schedule
• Sampling Efficiency

Innovation

• Matrix-Valued Diffusion Models
• Forward-Evolution Error Analysis
• Adaptive Grid Design
• KL Divergence Complexity
• Variance-Preserving Diffusions