Projected and Reflected Sliced Wasserstein Kernels for Persistence Diagrams

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.