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 $d_{\mathrm{SK}}$, 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 $O(N\log N)$ 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 $d_{\mathrm{SK}}$ 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 $d_{\mathrm{SK}}$ 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 $d_{\mathrm{SK}}$, 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 $d_{\mathrm{SK}}$ kernel supports other kernel-based analysis tasks, as illustrated by its use for contiguous segmentation of ordered diagram collections in our experiments.