2022/01/30 by Han, Andi, Mishra, Bamdev, Jawanpuria, Pratik +1
#FOS: Computer and information sciences #FOS: Mathematics #Functional Analysis (math.FA) #Machine Learning (stat.ML) #Optimization and Control (math.OC)
paper · doi:10.48550/arxiv.2201.12933
In this work, we study the optimal transport (OT) problem between symmetric positive definite (SPD) matrix-valued measures. We formulate the above as a generalized optimal transport problem where the cost, the marginals, and the coupling are represented as block matrices and each component block is a SPD matrix. The summation of row blocks and column blocks in the coupling matrix are constrained by the given block-SPD marginals. We endow the set of such block-coupling matrices with a novel Riemannian manifold structure. This allows to exploit the versatile Riemannian optimization framework to solve generic SPD matrix-valued OT problems. We illustrate the usefulness of the proposed approach in several applications.