The Discriminability Criterion: Noise Schedule Design for Diffusion in Two-Mode Gaussian Mixtures

The current analytical optimal noise schedule for annealed Langevin dynamics diffusion is geometric and derived using an overlap criterion, under the idealization of data as a single point. As a first next step for generalization, we extend this work to a symmetric two-mode Gaussian mixture, decomposed into independent parallel and perpendicular components. We propose the discriminability criterion, which bounds cross-mode contamination in the diffusion process, and use it to identify three stages of the noise schedule, in which mode discrimination is unattainable, attainable, or already resolved. The coarsest stage is implemented as a single jump, motivated by the per-jump compute cost being concave in log-step-size. Further, we give a closed-form, non-asymptotic Langevin step budget for shape recovery (perpendicular direction). For mode identity recovery (parallel direction), we give a typical-case estimate, using an approximation that sharpens away from the ridge and is asymptotically exact far from it, with a Kramers-based worst-case estimate valid in the large-barrier regime $\Delta V(\delta) \gg 1$ (derived using a mean-field potential where bifurcation follows the Curie–Weiss self-consistency equation). Our two-mode analysis is intended to be a tractable first step toward analytical and mechanism-level understanding of the role of data structure in discrete diffusion dynamics and how it influences noise schedule design.