Characterizing stratified structure from volume-growth laws for noisy point clouds

Develop a method to precisely characterize the stratified space structure of a noisy point-cloud sample using volume-growth laws—such as the Volume Growth Transform—including identification of strata, their local dimensions, and intersection structure from observed log–volume growth behavior.

Background

The authors reject the manifold and fiber-bundle hypotheses for the agent’s latent space by analyzing volume-growth behavior and local dimension distributions, arguing for a stratified-space model. While general analytic formulas for stratified-space volume growth involve Lipschitz–Killing measures and are difficult to estimate, they provide a realization theorem showing that a broad class of piecewise-linear volume-growth curves can be modeled by stratified spaces.

However, they note a methodological gap: there is currently no procedure to recover or precisely characterize the stratified structure of a noisy point-cloud sample directly from volume-growth laws. Filling this gap would enable rigorous identification of strata and their relationships from embeddings learned by transformer models.