Sierpiński--Knopp Wasserstein Distance for Persistence Diagrams and Applications to 2-Wasserstein Approximation
This paper introduces the Sierpiński-Knopp (SK) Wasserstein distance, a fast metric between persistence diagrams. The SK-Wasserstein distance, denoted d_{SK}, maps diagram points and their diagonal projections to the unit interval via the Sierpiński-Knopp space-filling curve on the upper diagonal triangle. The encoded point sets are then efficiently matched via one-dimensional optimal assignment, in (O(NN)) steps, yielding an explicit diagonal-aware point assignment between the two input persistence diagrams. We show that the SK-Wasserstein distance controls the classical (2)-Wasserstein distance between diagrams, admits an explicit isometric embedding into a Hilbert space, and induces a positive-definite Gaussian kernel, making the resulting geometry directly compatible with Euclidean and kernel-based learning methods. A tighter surrogate dissimilarity, noted (W_Γ), is also introduced based on the point assignments along the curve. Experiments on 12 scientific collections comprising 227 diagrams show median per-collection speedup of (d_{SK}) over state-of-the-art approximations of (W_2) is (626), while the aggregate speedup over the full benchmark is (2100). Average-linkage partitions obtained from (d_{SK}) and (W_Γ) each exactly match the corresponding (W_2) partition on 8 of the 12 collections. Hilbert (k)-means and Gaussian spectral clustering, both based on (d_{SK}), achieve mean adjusted Rand indices (ARI) of (0.756) and (0.800), respectively, with respect to the benchmark reference partitions, compared to (0.750) obtained by average linkage on (W_2). The Gaussian (d_{SK}) kernel supports other kernel-based analysis tasks, as illustrated by its use for contiguous segmentation of ordered diagram collections in our experiments.