The authors propose a new way to measure whether a high-dimensional distribution is fragmented into separated clusters ("mode separation") versus just spread out. They construct a diffusion process whose stationary distribution matches the data, then analyze its autocovariance: SSA (a scalar summarizing barrier strength) and DA (directions ordered by metastability, not variance). They derive theory under a Gaussian null and apply the method using pretrained score-based generative models to scale to high dimensions.
Main takeaways:
- Mode separation (how sharply a distribution splits into barrier-separated clusters) is geometrically distinct from dispersion, but existing tools like entropy and PCA don't capture it
- They use a reversible diffusion process with the target density as its equilibrium and extract two readouts from its autocovariance matrix
- SSA (Sum of Squared Autocorrelations) is a scalar that rises with barrier strength; DA (Dominant Autocorrelation directions) finds metastable directions instead of high-variance ones like PCA
- The method works with samples and a score function, so it scales via pretrained diffusion models
- Applications include Gaussian mixtures (SSA tracks mutual information), SDXL image generation (reveals structure entropy/PCA miss), and molecular dynamics (recovers known slow degrees of freedom)