Reproduce “Statistical learning on measures: an application to persistence diagrams” on OpenResearch

About this paper

Olympio Hacquard, Gilles Blanchard, Clément Levrard

A statistical learning framework is introduced for supervised binary classification when inputs are general measures, rather than traditional vectors. This approach provides generalization guarantees and empirically demonstrates that features often considered "topological noise" in persistence diagrams contain valuable information for classification across various datasets.

Key insights

• The generalization ability of classifiers operating on measures is directly linked to and controlled by the complexity of simpler base classifiers defined on the underlying data space.
• Features traditionally regarded as "topological noise" (low-persistence features) in persistence diagrams are not just artifacts but carry significant, discriminative information about sampling densities and other fine-grained data properties.
• Simple, interpretable region-based classifiers, when aggregated via boosting, can effectively discriminate between classes in measure spaces, providing inherent explainability regarding which regions are important.

Summary by alphaXiv.