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Optimal Transport in Reproducing Kernel Hilbert Spaces: Theory and Applications

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In this paper, we present a mathematical and computational framework for comparing and matching distributions in reproducing kernel Hilbert spaces (RKHS). This framework, called optimal transport in RKHS, is a… Click to show full abstract

In this paper, we present a mathematical and computational framework for comparing and matching distributions in reproducing kernel Hilbert spaces (RKHS). This framework, called optimal transport in RKHS, is a generalization of the optimal transport problem in input spaces to (potentially) infinite-dimensional feature spaces. We provide a computable formulation of Kantorovich's optimal transport in RKHS. In particular, we explore the case in which data distributions in RKHS are Gaussian, obtaining closed-form expressions of both the estimated Wasserstein distance and optimal transport map via kernel matrices. Based on these expressions, we generalize the Bures metric on covariance matrices to infinite-dimensional settings, providing a new metric between covariance operators. Moreover, we extend the correlation alignment problem to Hilbert spaces, giving a new strategy for matching distributions in RKHS. Empirically, we apply the derived formulas under the Gaussianity assumption to image classification and domain adaptation. In both tasks, our algorithms yield state-of-the-art performances, demonstrating the effectiveness and potential of our framework.

Keywords: transport; hilbert spaces; reproducing kernel; transport reproducing; optimal transport; kernel hilbert

Journal Title: IEEE Transactions on Pattern Analysis and Machine Intelligence
Year Published: 2020

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