Certified Topological Interaction in Neural Representations: Exact Tests and the Statistic They Require

2026-09-25T08:12:34.207Z

• title: Certified Topological Interaction in Neural Representations: Class Disentanglement Is Mostly Pairwise → Certified Topological Interaction in Neural Representations: Exact Tests and the Statistic They Require
• summary: Class disentanglement (the separation of a representation's class-conditional point clouds along depth and over training) is usually read off descriptive curves. We measure it as certified topological interaction between labeled point clouds, using the recently introduced Intersection Euler Characteristic Profile: the Euler characteristic of the overlap of the clouds' ball unions as a function of scale, computed by one Alpha-complex sweep with no boundary-matrix reduction. Every number carries a test: exact permutation tests in both directions, a guarded separation certificate, and a paired te → Class disentanglement--the separation of a representation's class-conditional point clouds along depth and over training--is measured by descriptive curves: the sentence such a study wants to write, layer l+1 is more disentangled than layer l, is an eyeball judgement with no null. We supply the inferential layer for a topological measurement of class overlap, the Intersection Euler Characteristic Profile: the Euler characteristic of the overlap of the clouds' ball unions as a function of scale, from one Alpha-complex sweep with no boundary-matrix reduction. Every number carries a test--exact p