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Quantum entropic regularization of matrix-valued optimal transport

Published online by Cambridge University Press:  28 September 2017

GABRIEL PEYRÉ ,
LÉNAÏC CHIZAT ,
FRANÇOIS-XAVIER VIALARD  and
JUSTIN SOLOMON
GABRIEL PEYRÉ
Affiliation:
CNRS and DMA, École Normale Supérieure, 45 rue d'Ulm - F 75230PARIScedex 05 email: gabriel.peyre@ens.fr
LÉNAÏC CHIZAT
Affiliation:
Ceremade, Univ. Paris-Dauphine and INRIA Mokaplan, Place du Maréchal de Lattre de Tassigny, 75016 Paris email: chizat@ceremade.dauphine.fr, vialard@ceremade.dauphine.fr
FRANÇOIS-XAVIER VIALARD
Affiliation:
Ceremade, Univ. Paris-Dauphine and INRIA Mokaplan, Place du Maréchal de Lattre de Tassigny, 75016 Paris email: chizat@ceremade.dauphine.fr, vialard@ceremade.dauphine.fr
JUSTIN SOLOMON
Affiliation:
EECS and CSAIL, MIT, 32 Vassar Street, room 32-D460 Cambridge, MA 02139 email: jsolomon@mit.edu
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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 optimal transport (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 barycentres 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.

Keywords

Optimal transporttensor fieldPSD matricesquantum entropy
Type
Papers
Information
European Journal of Applied Mathematics , Volume 30 , Special Issue 6: Applied Optimal Transport , December 2019 , pp. 1079 - 1102
Copyright
© Cambridge University Press 2017 

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Footnotes

†

The work of Gabriel Peyré has been supported by the European Research Council (ERC project SIGMA-Vision). J. Solomon acknowledges support of Army Research Office grant W911NF-12-R-0011 (“Smooth Modeling of Flows on Graphs”).

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