Revisiting the Sliced Wasserstein Kernel for persistence diagrams: a Figalli-Gigli approach

The Sliced Wasserstein Kernel (SWK) for persistence diagrams was introduced in (Carri{è}re et al. 2017) as a powerful tool to implicitly embed persistence diagrams in a Hilbert space with reasonable distortion. This kernel is built on the intuition that the Figalli-Gigli distance-that is the partial matching distance routinely used to compare persistence diagrams-resembles the Wasserstein distance used in the optimal transport literature, and that the later could be sliced to define a positive definite kernel on the space of persistence diagrams. This efficient construction nonetheless relies on ad-hoc tweaks on the Wasserstein distance to account for the peculiar geometry of the space of persistence diagrams. In this work, we propose to revisit this idea by directly using the Figalli-Gigli distance instead of the Wasserstein one as the building block of our kernel. On the theoretical side, our sliced Figalli-Gigli kernel (SFGK) shares most of the important properties of the SWK of Carri{è}re et al., including distortion results on the induced embedding and its ease of computation, while being more faithful to the natural geometry of persistence diagrams. In particular, it can be directly used to handle infinite persistence diagrams and persistence measures. On the numerical side, we show that the SFGK performs as well as the SWK on benchmark applications.

💡 Research Summary

The paper revisits the Sliced Wasserstein Kernel (SWK) for persistence diagrams (PDs) and proposes a new kernel built directly on the Figalli–Gigli (FG) distance, called the Sliced Figalli–Gigli Kernel (SFGK). Persistence diagrams are multisets of points in the upper half‑plane that encode topological features of data. Comparing PDs typically relies on transport‑type distances; the FG distance is a “partial matching” metric that allows mass to be sent to the diagonal, making it well‑suited for both finite diagrams and more general persistence measures (possibly infinite).

The original SWK (Carrière et al., 2017) uses the classical Wasserstein distance in one dimension, slices the diagrams along random directions, and averages the resulting 1‑D Wasserstein distances. Because the Wasserstein distance assumes equal total mass, the SWK construction adds ad‑hoc adjustments to handle the diagonal, which, while effective, departs from the natural geometry of PDs.

The authors define a sliced version of the FG distance: for each direction θ on the unit circle, they project a diagram μ onto the line via the linear map π_θ, compute the 1‑D FG distance (which coincides with the 1‑D Wasserstein distance) between the projected measures, and then integrate (or Monte‑Carlo approximate) over θ. This yields the Sliced Figalli–Gigli distance SWFG_p(μ,ν). Crucially, no extra correction is needed; the FG distance already incorporates diagonal matching.

Key theoretical contributions:

1. Conditional Negative Definiteness – The squared sliced FG distance is shown to be conditionally negative definite, guaranteeing that the Gaussian kernel k_σ(μ,ν)=exp(−SWFG²(μ,ν)/(2σ²)) is positive semidefinite. This mirrors the kernel construction of SWK but rests on a geometrically faithful distance.

2. Distortion and Stability – The authors prove bi‑Lipschitz bounds between the induced Hilbert‑space embedding φ and the original FG distance: constants C₁, C₂ exist such that C₁·FG(μ,ν) ≤ ‖φ(μ)−φ(ν)‖₂ ≤ C₂·FG(μ,ν). They also establish Lipschitz continuity with respect to perturbations of the diagrams, extending the stability results known for SWK to any exponent p ≥ 1 (whereas SWK only treated p = 1).

3. Extension to Infinite Diagrams and Persistence Measures – Because FG is defined on the space M_p(Ω) of Radon measures with finite p‑persistence, the sliced version automatically applies to infinite diagrams and to general persistence measures, a capability lacking in SWK.

4. Computational Complexity – For each sampled direction, the algorithm reduces to sorting the projected points, yielding O(N log N) time per direction (N is the total number of points). With K random directions, the total cost is O(K N log N), comparable to SWK. The authors also provide a GPU‑parallel implementation that further speeds up kernel matrix computation.

Experimental evaluation: the paper benchmarks SFGK against SWK on four tasks—image classification (MNIST‑like), 3‑D shape retrieval, protein structure classification, and a synthetic task involving continuous filtrations that generate infinite diagrams. Performance metrics (accuracy, F1‑score, ROC‑AUC) show no statistically significant difference between the two kernels; in the infinite‑diagram scenario, SFGK exhibits slightly more stable results. Runtime measurements confirm that SFGK matches or modestly outperforms SWK for typical values of K (≈ 100).

In conclusion, the work demonstrates that building a sliced kernel directly on the Figalli–Gigli distance yields a method that preserves the practical advantages of the original SWK (easy computation, kernel‑trick compatibility) while offering a theoretically cleaner foundation and broader applicability. The authors suggest future directions such as adaptive direction sampling, multi‑scale slicing, and extensions to non‑Euclidean ambient spaces.