Sierpi'nski--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(N N) 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 times, while the aggregate speedup over the full benchmark is 2100 times. 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.