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

TL;DR

The paper addresses kernel-based learning with persistence diagrams by revisiting the distance underpinning the Sliced Wasserstein Kernel and replacing the Wasserstein-based slice with the Figalli–Gigli distance, yielding the Sliced Figalli–Gigli distance (SFG_p). This approach preserves the strong geometric alignment with persistence diagrams, extends naturally to infinite diagrams and persistence measures, and comes with stability guarantees and efficient computation. The authors show that the induced SFG_p kernel is positive definite (for p ∈ [1,2]) and topologically equivalent to FG_p, enabling Hilbert-space embeddings while retaining the intrinsic PD geometry. Empirically, SFGK performs on par with SWK on benchmark tasks such as orbit recognition and texture classification, while offering a broader, measure-theoretic framework and potential for generalizations. Overall, the work provides a theoretically sound and practically viable refinement of sliced distances for persistence diagrams with broader applicability and robustness to diagram size.

Abstract

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.

Figures (9)

• Figure 1: Projections of two measures $\mu$ (circles) and $\nu$ (crosses) and their respective projections on the geodesic with parameter $t$. Points outside of the region delimited by $x_1 = t$ and $x_2 = t$ (which we denote $\Omega_t$) are projected onto $\partial \Omega$. As $t$ describes $\mathbb{R}$, $\Omega_t$slides across the upper half-plane and captures information about different parts of the diagram. Observe also that a point $x$ belongs to $\Omega_t$ for $t$ belonging to an interval of length proportional to $d(x,\partial \Omega)$, hence the need for the renormalization we introduce.
• Figure 2: Change of coordinates used in the proof of \ref{['lem:injectivity-transform']}.
• Figure 3: $B_{\varepsilon/2}$ (blue) and $B_\varepsilon$ (red)
• Figure 4: Boxplot of the ratio between the approximation of $\mathrm{SFG}$ and its exact value with respect to the number of samples used for the approximation. The persistence diagrams we used to get this plot were taken from the Orbits dataset we generated (see \ref{['subs:experimental-results']}). The uniform sampling leads to better convergence speeds while the gaussian KDE sampling tends to underestimate the value of $\mathrm{SFG}$.
• Figure 5: Boxplot of the ratio between $\mathrm{SFG}$ and the theoretical bound of \ref{['thm:ineg-projn']} for different values of $p$. For each value of $p$, the boxplot was obtained with 500 persistence diagrams which were sampled uniformally for (a) and using the procedures described in \ref{['subs:experimental-results']} for (b) and (c).

Theorems & Definitions (69)

• Remark 1
• Remark 2
• Remark 3
• Proposition 1: Geodesics in $\tilde{\Omega}$
• Proposition 2: Projections on geodesics
• proof
• Lemma 1
• proof
• Remark 4
• Definition 1: The Sliced Figalli--Gigli distance