A View on Optimal Transport from Noncommutative Geometry

Abstract

We discuss the relation between the Wasserstein distance of order 1 between probability distributions on a metric space, arising in the study of Monge-Kantorovich transport problem, and the spectral distance of noncommutative geometry. Starting from a remark of Rieffel on compact manifolds, we first show that on any - i.e. non-necessary compact - complete Riemannian spin manifolds, the two distances coincide. Then, on convex manifolds in the sense of Nash embedding, we provide some natural upper and lower bounds to the distance between any two probability distributions. Specializing to the Euclidean space $R^n$, we explicitly compute the distance for a particular class of distributions generalizing Gaussian wave packet. Finally we explore the analogy between the spectral and the Wasserstein distances in the noncommutative case, focusing on the standard model and the Moyal plane. In particular we point out that in the two-sheet space of the standard model, an optimal-transport interpretation of the metric requires a cost function that does not vanish on the diagonal. The latest is similar to the cost function occurring in the relativistic heat equation.

Enhanced version: a new section with noncommutative examples has been added. In the commutative case, the completeness condition is underlined. The geodesic aspect of the distance, as well as alternative definitions, are discussed

References in corpus (2)

• The spectral distance on the Moyal plane
• On the distance observable in the Moyal plane and in a novel two-dimensional space with string-theory pregeometry

Cited by in corpus (14)

• Spectral truncations in noncommutative geometry and operator systems
• The spectral distance on the Moyal plane
• Emergence of the Circle in a Statistical Model of Random Cubic Graphs
• Spectral Distances: Results for Moyal Plane and Noncommutative Torus
• On the pseudo-manifold of quantum states
• Length and distance on a quantum space
• A dual formula for the spectral distance in noncommutative geometry
• Connes distance and optimal transport
• Noncommutative geometry of the Moyal plane: translation isometries, Connes' distance on coherent states, Pythagoras equality
• Deformations of the Canonical Commutation Relations and Metric Structures
• Metrics for matrix-valued measures via test functions
• $L^p$-Wasserstein distances on state and quasi-state spaces of $C^{\ast }$-algebras
• A critical survey of twisted spectral triples beyond the Standard Model
• Towards a Monge-Kantorovich metric in noncommutative geometry