← 研踪 PaperTrailarXiv:2605.08777 · v2
Measuring and Decomposing Mode Separation via the Canonical Diffusion
首次提交 2026-05-09 16:05
版本更新 2026-11-06 13:49
雷达观测 2026-11-09 11:19
摘要
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. Modes are often the meaningful alternatives in the data, such as the conformations of a molecule or the distinct semantic types among a generative model's outputs. Mode separation is distinct from dispersion, which is what standard label-free tools capture: differential entropy rises with spread regardless of fragmentation, and PCA orders directions by variance regardless of barriers. Mutual information does capture it, but only given the latent component labels, which are usually unavailable. We measure mode separation through a single stochastic process intrinsic to the density, namely the unique reversible diffusion with $f$ as its stationary distribution and a constant scalar diffusion coefficient. From its autocovariance matrix we extract two readouts. The first, SSA (Sum of Squared Autocorrelations), is a scalar barrier-sensitive measure. The second, DA (Dominant Autocorrelation directions), gives linear projections ordered by metastability rather than variance. Under an isotropic-Gaussian null, we derive the limiting spectrum of the empirical autocovariance in closed form, generalizing the Marchenko--Pastur law. Its upper edge gives a noise floor that separates metastable directions from sampling noise. 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.
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v2 · 2026-11-06 13:49 · 最新观测版本
Measuring and Decomposing Mode Separation via the Canonical Diffusion
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. Modes are often the meaningful alternatives in the data, such as the conformations of a molecule or the distinct semantic types among a generative model's outputs. Mode separation is distinct from dispersion, which is what standard label-free tools capture: differential entropy rises with spread regardless of fragmentation, and PCA orders directions by variance regardless of barriers. Mutual information does capture it, but only given the latent component labels, which are usually unavailable. We measure mode separation through a single stochastic process intrinsic to the density, namely the unique reversible diffusion with $f$ as its stationary distribution and a constant scalar diffusion coefficient. From its autocovariance matrix we extract two readouts. The first, SSA (Sum of Squared Autocorrelations), is a scalar barrier-sensitive measure. The second, DA (Dominant Autocorrelation directions), gives linear projections ordered by metastability rather than variance. Under an isotropic-Gaussian null, we derive the limiting spectrum of the empirical autocovariance in closed form, generalizing the Marchenko--Pastur law. Its upper edge gives a noise floor that separates metastable directions from sampling noise. 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.
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