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 SW1, which is conditionally negative definite (CND). We study the projected (SWp) and reflected (RWp) Sliced Wasserstein distances for every order p≥1, and compare them with the p-Wasserstein distance dp and with a reflected variant dpr, in which projection onto the diagonal is replaced by reflection across it. We show that 21/pdp≤ dpr≤ 2dp with sharp constants, so that d1r=2d1 and dpr inherits the stability of dp, although it is a metric only for p=1. The distances SW1 and RW1 are CND metrics, and RW1 gives rise to a new positive definite kernel, stable with respect to d1. For p>1, neither SWp nor RWp satisfies the triangle inequality, and rigorous interval-arithmetic certificates prove that SW2 and RW2 are not CND; numerical experiments suggest that the CND property fails for every p above some p*∈(1,1.2]. On diagrams with at most N points, SWp and RWp are metrically equivalent to dp and dpr, with sharp upper constants. On an orbit recognition benchmark, the kernels built from SW1 and RW1 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 SW2, which is not positive definite.
Related
- Persistence diagrams of finite topological spaces
- Projected Wasserstein-Fisher-Rao Flows for Hawkes Background-Intensity Estimation
- A result of continuous stability for generalized persistence diagrams
- On the Spectral Synthesis of Lipschitz Vectorizations of Persistence Diagrams
- A Jump-Modified Wasserstein Gradient Flow for Maximum Mean Discrepancy with Energy Kernels
BibTeX
@misc{castaneda2026projected,
title = {{Projected and Reflected Sliced Wasserstein Kernels for Persistence Diagrams}},
author = {Julián Uribe Castañeda and Mircea Petrache},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.06619},
eprint = {2611.06619},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.06619},
}
RIS
TY - UNPB
TI - Projected and Reflected Sliced Wasserstein Kernels for Persistence Diagrams
AU - Julián Uribe Castañeda
AU - Mircea Petrache
PY - 2026
DA - 2026/11/04
PB - arXiv
DO - 10.48550/arxiv.2611.06619
UR - https://arxiv.org/abs/2611.06619
AB - 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 SW_1, which is conditionally negative definite (CND). We study the projected (SW_p) and reflected (RW_p) Sliced Wasserstein distances for every order p≥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/pd_p≤ d_p^r≤ 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 SW_1 and RW_1 are CND metrics, and RW_1 gives rise to a new positive definite kernel, stable with respect to d_1. For p>1, neither SW_p nor RW_p satisfies the triangle inequality, and rigorous interval-arithmetic certificates prove that SW_2 and RW_2 are not CND; numerical experiments suggest that the CND property fails for every p above some p^*∈(1,1.2]. On diagrams with at most N points, SW_p and 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 SW_1 and 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 SW_2, which is not positive definite.
ER -