Projected and Reflected Sliced Wasserstein Kernels for Persistence Diagrams

Abstract

Persistence diagrams do not naturally live in a Hilbert space. Carrière, Cuturi, and Oudot embedded them in kernel methods through the Sliced Wasserstein kernel, built from the Sliced Wasserstein distance ${\mathrm{SW}}_1$, which is conditionally negative definite (CND). We study the projected (${\mathrm{SW}}_p$) and reflected (${\mathrm{RW}}_p$) Sliced Wasserstein distances for every order $p\ge 1$, and compare them with the $p$-Wasserstein distance $d_p$ and with a reflected variant $d_p^r$, in which projection onto the diagonal is replaced by reflection across it. We show that $2^{1/p}{d}_p\le d_p^r\le 2d_p$ with sharp constants, so that $d_1^r=2d_1$ and $d_p^r$ inherits the stability of $d_p$, although it is a metric only for $p=1$. The distances ${\mathrm{SW}}_1$ and ${\mathrm{RW}}_1$ are CND metrics, and ${\mathrm{RW}}_1$ gives rise to a new positive definite kernel, stable with respect to $d_1$. For $p>1$, neither ${\mathrm{SW}}_p$ nor ${\mathrm{RW}}_p$ satisfies the triangle inequality, and rigorous interval-arithmetic certificates prove that ${\mathrm{SW}}_2$ and ${\mathrm{RW}}_2$ are not CND; numerical experiments suggest that the CND property fails for every $p$ above some $p^{\ast }\in (1,1.2]$. On diagrams with at most $N$ points, ${\mathrm{SW}}_p$ and ${\mathrm{RW}}_p$ are metrically equivalent to $d_p$ and $d_p^r$, with sharp upper constants. On an orbit recognition benchmark, the kernels built from ${\mathrm{SW}}_1$ and ${\mathrm{RW}}_1$ reach about 85% accuracy, outperform the persistence scale space kernel, perform on par with the persistence weighted Gaussian kernel, and are about one percentage point below the kernel built from ${\mathrm{SW}}_2$, which is not positive definite.

arXiv comment50 pages, 7 figures, 10 tables