Mathematical Approaches to Clustering
summary
Compares concrete algorithms and mathematical frameworks and gives clear takeaways: k-means fails on elongated strip data because it enforces convex Voronoi tiles, analyzes single linkage and shows a thresholded version is functorial and links this to density-aware variants using small simplices similar to DBSCAN, proves the crisp claim Kleinberg impossibility: no clustering rule satisfies scale consistency and richness simultaneously and points to persistent homology to detect loops and higher‑dimensional features.
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chapters (8)
- 0:00Introduction
- 1:54Metric Space
- 4:50Kmeans
- 14:27axioms for clustering
- 16:35Objections
- 21:42Categories
- 23:01Single Linkage
- 24:24Persistent Homology
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