Clustering, factor discovery and optimal transport

Abstract

The clustering problem, and more generally latent factor discovery or latent space inference, is formulated in terms of the Wasserstein barycenter problem from optimal transport. The objective proposed is the maximization of the variability attributable to class, further characterized as the minimization of the variance of the Wasserstein barycenter. Existing theory, which constrains the transport maps to rigid translations, is extended to affine transformations. The resulting non-parametric clustering algorithms include $k$-means as a special case and exhibit more robust performance. A continuous version of these algorithms discovers continuous latent variables and generalizes principal curves. The strength of these algorithms is demonstrated by tests on both artificial and real-world data sets.

Discriminative Radial Domain Adaptation

“…Further, for relaxing the objective and stablizing the end-to-end training we use entropic optimal transport [51] distance defined by: (8) where µ = n i=1 δ xi the measure of data instances and µ a = k j=1 δ aj the measure of anchors, d(•, •) is euclidean distance metric, π is the transport plan, Π(•, •) represent set of total transport permutation combination, ≥ 0 is the regularization coefficient. As a relevant metric for assigning samples to the best-fitted anchors, optimal transport distance can lead to a more reliable assignment than nearest neighbor search [52]. Therefore, by optimizing G θ (•) in minimizing L ot = OT θ (X s , N s )+OT θ (X t , N t ) in both source and target domain independently, the extracted features in both domains are more compactly arranged around their radial structures, the structure faithfulness requirement can be indirectly achieved.…”