Persistent-Homology

Disclosure: this post was written with the assistance of an AI system (Claude), which wrote the analysis code, ran the experiments and drafted the text. The topic, the constraints, the data choices and the final review are the author’s. Persistent homology arrives with a large vocabulary and, in dimension zero, a very small computation: walk the edges shortest first and record the length of each one that joins two components. That is the single-linkage dendrogram’s merge heights, and not approximately — the two return arrays that are equal float for float, on four designs from 12 to 60 points and 1 to 30 dimensions....

Disclosure: this post was written with the assistance of an AI system (Claude), which wrote the analysis code, ran the experiments and drafted the text. The topic, the constraints, the data choices and the final review are the author’s. Three measurements, and the third one reversed the plan. First: the H0 death spread, as a fraction of its own mean, falls from 4.87 at two dimensions to 0.078 at 768 on pure noise....