Scaling Optimal Transport to High-Dimensional Gaussian Distributions with Application to Domain Adaptation

Optimal transport supports clustering, dictionary learning and domain adaptation, but high-dimensional data often force projections that discard information. Charles Bouveyron develops an alternative for Gaussian distributions with parsimonious covariance structure and low intrinsic dimension. A low-dimensional decomposition of empirical covariance matrices gives an efficient, robust closed form for the 2-Wasserstein distance without projecting the data. An explicit Monge map uses exact covariance square roots and inverse square roots. Numerical comparisons demonstrate analytical and computational benefits, including competitiveness with model-free approaches for non-Gaussian data and applications to unsupervised domain adaptation for classification.