黎曼块SPD耦合流形及其在最优输运中的应用
Riemannian block SPD coupling manifold and its application to optimal transport
摘要：
本文研究了对称正定矩阵值测度之间的最优输运问题。我们将上述问题公式化为广义最优输运问题，其中成本、边际和耦合表示为块矩阵，并且每个组成块是SPD矩阵。耦合矩阵中的行块和列块的总和受给定的块-SPD边缘的约束。我们赋予这样的块耦合矩阵的集合一个新的黎曼流形结构。这允许利用多功能的黎曼优化框架来解决一般的SPD矩阵值OT问题。我们说明了所提出的方法在几个应用程序中的实用性。
英文摘要：
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.
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