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arXiv · 1612.08731

Quantum Optimal Transport for Tensor Field Processing

Gabriel Peyré·Lenaïc Chizat·François-Xavier Vialard·Justin Solomon
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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.

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Gabriel Peyré, Lenaïc Chizat, François-Xavier Vialard, Justin Solomon. 2016-12-20. Quantum Optimal Transport for Tensor Field Processing. https://doi.org/10.1017/s0956792517000274

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