Rigidity and Quantitative Stability of the Sliced Wasserstein Deficit

Abstract

The sliced Wasserstein distance $SW_2(μ,ν)$ compares high-dimensional probability measures by averaging one-dimensional optimal transport distances over linear projections. Although sliced Wasserstein distances are now standard computational tools in statistics, imaging, and machine learning, the rigidity behind the elementary comparison \[ SW_2^2(μ,ν)\leq \frac1d W_2^2(μ,ν) \] has not been systematically studied. Let $μ,ν\in\mathcal P_2(\mathbb R^d)$, $d\ge2$, with $μ\ll\mathcal L^d$, and define the sliced Wasserstein deficit by \[ {\mathrm D}(μ,ν):=\frac1d W_2^2(μ,ν)-SW_2^2(μ,ν)\geq 0. \] We prove that ${\mathrm D}(μ,ν)=0$ if and only if the Brenier map $T=\nablaφ$ from $μ$ to $ν$ is homothetic affine, \[ T(x)=λx+b \qquad μ\text{-a.e.}, \] for some $λ\ge0$ and $b\in \mathbb R^d$. For quantitative stability, we introduce the sliced Poincaré--Korn (SPK) constant $κ_{\mathrm{SPK}}(μ)$, defined as an new spectral gap of an averaged ridge-projection quadratic form on gradient fields modulo the family $\{λx+b\}$. Whenever this constant is positive, we prove a stability estimate for the sliced Wasserstein deficit, up to a one-dimensional Lipschitz scale for the projected monotone transports. We obtain the sharp SPK constant for the Gaussian measures as the most important example, and establish positive SPK bounds for bounded perturbations of the Gaussian and compact classes of gradient fields for fixed source measures. Finally, we show that anisotropic Gaussians give a sharp obstruction: neither a Bakry--Émery lower curvature bound nor a usual Poincaré inequality alone can imply a global sliced Poincaré--Korn inequality.