Optimal Transport in Reproducing Kernel Hilbert Spaces: Theory and Applications

Area of Science:

• Machine Learning
• Mathematical Analysis
• Statistics

Background:

• Comparing and matching probability distributions is crucial in machine learning.
• Existing optimal transport methods are often limited to finite-dimensional input spaces.
• Reproducing kernel Hilbert spaces (RKHS) offer powerful tools for handling complex data features.

Purpose of the Study:

• To develop a general mathematical and computational framework for optimal transport in RKHS.
• To generalize the optimal transport problem to potentially infinite-dimensional feature spaces.
• To provide computable formulations and practical applications of optimal transport in RKHS.

Main Methods:

• Developed a computable formulation of Kantorovich's optimal transport in RKHS.
• Derived closed-form expressions for Wasserstein distance and optimal transport maps for Gaussian distributions in RKHS using kernel matrices.
• Generalized the Bures metric and extended the correlation alignment problem to Hilbert spaces.

Main Results:

• Obtained closed-form solutions for optimal transport in Gaussian RKHS.
• Introduced a novel metric for covariance operators in infinite dimensions.
• Proposed a new distribution matching strategy for RKHS.
• Achieved state-of-the-art performance in image classification and domain adaptation tasks.

Conclusions:

• The proposed optimal transport in RKHS framework is effective for comparing and matching distributions.
• The derived formulas provide efficient and accurate solutions for Gaussian distributions.
• The framework demonstrates significant potential for advancing machine learning applications like domain adaptation and image classification.