Provably Correct k-Means Clustering of Persistence Diagrams

We study $k$-means clustering of persistence diagrams through their {Hilbert}-space embeddings, focusing primarily on persistence landscapes. Although persistence landscapes are widely used in practice, it remains unclear when clustering them faithfully reflects clustering in the diagram space. We give simple, verifiable geometric conditions under which (i) nearest-center labels in the landscape space exactly agree with those in the diagram space, and (ii) the landscape $k$-means objective provably calibrates the diagram-space objective, leveraging tools from modern statistical learning theory. Combined with known fast rates for $k$-means in {Hilbert} spaces, our results show that landscape-based $k$-means provides a statistically and computationally efficient surrogate for diagram-space $k$-means with explicit performance guarantees and a controllable geometric distortion.