An Efficient Algorithm for Matrix-Valued and Vector-Valued Optimal Mass Transport
Preprint
Abstract
We present an efficient algorithm for recent generalizations of optimal mass transport theory to matrix-valued and vector-valued densities. These generalizations lead to several applications including diffusion tensor imaging, color images processing, and multi-modality imaging. The algorithm is based on sequential quadratic programming (SQP). By approximating the Hessian of the cost and solving each iteration in an inexact manner, we are able to solve each iteration with relatively low cost while still maintaining a fast convergent rate. The core of the algorithm is solving a weighted Poisson equation, where different efficient preconditioners may be employed. We utilize incomplete Cholesky factorization, which yields an efficient and straightforward solver for our problem. Several illustrative examples are presented for both the matrix and vector-valued cases.Keywords
All Related Versions
This publication has 18 references indexed in Scilit:
- Matrix Optimal Mass Transport: A Quantum Mechanical ApproachIEEE Transactions on Automatic Control, 2017
- Gradient flow and entropy inequalities for quantum Markov semigroups with detailed balanceJournal of Functional Analysis, 2017
- Joint hydrogeophysical inversion: state estimation for seawater intrusion models in 3DComputational Geosciences, 2016
- Iterative Bregman Projections for Regularized Transportation ProblemsSIAM Journal on Scientific Computing, 2015
- Numerical solution of the Optimal Transportation problem using the Monge–Ampère equationJournal of Computational Physics, 2014
- An Inexact SQP Method for Equality Constrained OptimizationSIAM Journal on Optimization, 2008
- Generalization of an Inequality by Talagrand and Links with the Logarithmic Sobolev InequalityJournal of Functional Analysis, 2000
- A computational fluid mechanics solution to the Monge-Kantorovich mass transfer problemNumerische Mathematik, 2000
- Polar factorization and monotone rearrangement of vector‐valued functionsCommunications on Pure and Applied Mathematics, 1991
- The incomplete Cholesky—conjugate gradient method for the iterative solution of systems of linear equationsJournal of Computational Physics, 1978