EGCA: A Spectral Perspective on Forward Process Design in Diffusion Models

The design of the forward diffusion process is a foundational yet under-theorized aspect of diffusion modeling. Although it determines how data are corrupted and how quickly the forward dynamics approach a reference distribution, forward-process design remains largely governed by canonical defaults rather than a unified analytical framework. We propose Eigenvalue-Guided Control and Analysis (EGCA) for diffusion models, a spectral framework that uses the principal eigenvalue of the infinitesimal generator to analyze and guide forward-process design. Under suitable ergodicity conditions, this eigenvalue governs the exponential convergence rate to stationarity, providing a compact and interpretable summary of forward dynamics. We establish sufficient uniqueness and ergodicity conditions, derive eigenvalue-based convergence bounds, and adapt a practical numerical estimator for the principal eigenvalue. Empirically, we show that the eigenvalue strongly tracks forward convergence speed and serves as an organizing coordinate for the efficiency--quality trade-off across tractable forward-generator families. Experiments across multiple image datasets and diverse diffusion settings, including training formulations, architectures, and samplers, demonstrate that eigenvalue-guided design maintains or improves generation quality while reducing training cost. Beyond a single generator choice, our framework offers a spectral perspective for understanding and comparing forward-process choices in diffusion models.