Sliced-Wasserstein Distances and Flows on Cartan-Hadamard Manifolds

TL;DR

This work derives general constructions of Sliced-Wasserstein distances on Cartan-Hadamard manifolds, Riemannian manifolds with non-positive curvature, which include among others Hyperbolic spaces or the space of Symmetric Positive Definite matrices.

Abstract

While many Machine Learning methods were developed or transposed on Riemannian manifolds to tackle data with known non Euclidean geometry, Optimal Transport (OT) methods on such spaces have not received much attention. The main OT tool on these spaces is the Wasserstein distance which suffers from a heavy computational burden. On Euclidean spaces, a popular alternative is the Sliced-Wasserstein distance, which leverages a closed-form solution of the Wasserstein distance in one dimension, but which is not readily available on manifolds. In this work, we derive general constructions of Sliced-Wasserstein distances on Cartan-Hadamard manifolds, Riemannian manifolds with non-positive curvature, which include among others Hyperbolic spaces or the space of Symmetric Positive Definite matrices. Then, we propose different applications. Additionally, we derive non-parametric schemes to minimize these new distances by approximating their Wasserstein gradient flows.

Figures (12)

• Figure 1: (Left) Orthogonal projection of points on a line passing through the origin 0. (Middle and Right) Illustration of the projection of 2d distributions on 3 different lines.
• Figure 2: Illustration of geodesics, of the tangent space and the exponential map on a Riemannian manifold.
• Figure 3: ( Left) On Euclidean spaces, the distance between the projections of two points belonging to a geodesic with the same direction is conserved. ( Middle) On Hyperbolic spaces, this is also the case using the horospherical projection as demonstrated in chami2021horopca, but not for geodesic projections ( Right).
• Figure 4: Illustration of the projection process of measures on geodesics $t\mapsto \exp_o(tv_1)$ and $t\mapsto \exp_o(tv_2)$.
• Figure 5: Projection of (red) points on a geodesic (black line) in the Poincaré ball and in the Lorentz model along Euclidean lines, geodesics or horospheres (in blue). Projected points on the geodesic are shown in green.

Theorems & Definitions (85)

• Definition 1: Gradient
• Proposition 1
• Proposition 2
• Proposition 3
• Definition 2: Cartan-Hadamard Sliced-Wasserstein
• Theorem 1: Pullback Euclidean Metric
• Proposition 4
• Definition 3: Mahalanobis Sliced-Wasserstein
• Proposition 5: Coordinate projections on Hyperbolic spaces
• Definition 4: Hyperbolic Sliced-Wasserstein