Quantum Optimal Transport for Tensor Field Processing

TL;DR

This paper introduces a quantum-inspired optimal transport framework for tensor fields, utilizing quantum entropy and a quantum Sinkhorn algorithm to efficiently compute transport plans and barycenters with applications in imaging and texture synthesis.

Contribution

It presents a novel quantum optimal transport formulation for tensor fields, including a regularized convex optimization approach and an extension of Sinkhorn algorithm to the quantum setting.

Findings

Efficient quantum Sinkhorn algorithm for tensor field transport

Successful application to imaging and texture synthesis tasks

New method generalizes classical OT to PSD matrix-valued measures

Abstract

This article introduces a new notion of optimal transport (OT) between tensor fields, which are measures whose values are positive semidefinite (PSD) matrices. This "quantum" formulation of OT (Q-OT) corresponds to a relaxed version of the classical Kantorovich transport problem, where the fidelity between the input PSD-valued measures is captured using the geometry of the Von-Neumann quantum entropy. We propose a quantum-entropic regularization of the resulting convex optimization problem, which can be solved efficiently using an iterative scaling algorithm. This method is a generalization of the celebrated Sinkhorn algorithm to the quantum setting of PSD matrices. We extend this formulation and the quantum Sinkhorn algorithm to compute barycenters within a collection of input tensor fields. We illustrate the usefulness of the proposed approach on applications to procedural noise generation, anisotropic meshing, diffusion tensor imaging and spectral texture synthesis.