Measuring and Decomposing Mode Separation via the Canonical Diffusion

2026-05-09

AI Summary

Characterizing mode separation—multimodal structures divided by energy barriers—in high-dimensional density distributions remains challenging for conventional measures such as differential entropy, PCA, or mutual information. This work proposes a novel approach based on reversible diffusion processes, leveraging only samples and the score function of the target density as its stationary distribution. Two key quantities are introduced via the autocovariance matrix: a scalar metric, Sum of Squared Autocorrelations (SSA), sensitive to barrier height, and Dominant Autocorrelation (DA) directions that yield linear projections ordered by metastability. Under an isotropic Gaussian null hypothesis, the authors derive a closed-form solution for the autocovariance spectrum, generalizing the Marchenko–Pastur law. Experiments demonstrate that SSA aligns with mutual information in Gaussian mixture models, uncovers structural features missed by PCA and entropy in SDXL image generation, and accurately recovers known slow dihedral angles in alanine dipeptide dynamics using only static samples.

Abstract

Mode separation, namely how sharply a distribution fragments into barrier-separated clusters, is a fundamental geometric property of densities, difficult to quantify in high dimensions. It is structurally distinct from dispersion, yet existing tools fall short: differential entropy rises with spread regardless of fragmentation, PCA orders directions by variance regardless of barriers, and mutual information requires a mixture decomposition one usually does not have. We measure mode separation through a single stochastic process intrinsic to the density: a unique reversible diffusion with $f$ as its stationary distribution and constant scalar diffusion coefficient. We extract two readouts from its autocovariance matrix: SSA (Sum of Squared Autocorrelations), a scalar barrier-sensitive measure; and DA (Dominant Autocorrelation directions), linear projections ordered by metastability rather than variance. Under an isotropic-Gaussian null, we derive a closed-form spectrum for the empirical autocovariance that generalizes Marchenko--Pastur, with an analytic upper edge that selects the lag at which DA is read off. Both readouts use only samples and a score function, scaling to high dimensions through pretrained score-based generative models via Tweedie's identity. We apply our framework to three settings: (i) synthetic Gaussian mixtures, where SSA tracks mutual information; (ii) SDXL text-to-image generations, where SSA and DA capture structure that entropy and PCA miss; and (iii) molecular dynamics of alanine dipeptide, where DA recovers the known slow backbone dihedrals from static samples alone.

Problem

• mode separation
• high-dimensional density
• barrier-separated clusters
• geometric property
• multimodal distribution

Innovation

• canonical diffusion
• mode separation
• autocovariance spectrum
• score-based generative models
• metastability