LCOT: Linear circular optimal transport
Rocio Diaz Martin Ivan Medri Yikun Bai Xinran Liu Kangbai Yan Gustavo K. Rohde Soheil Kolouri
The optimal transport problem for measures supported on non-Euclidean spaces
has recently gained ample interest in diverse applications involving
representation learning. In this paper, we focus on circular probability
measures, i.e., probability measures supported on the unit circle, and
introduce a new computationally efficient metric for these measures, denoted as
Linear Circular Optimal Transport (LCOT). The proposed metric comes with an
explicit linear embedding that allows one to apply Machine Learning (ML)
algorithms to the embedded measures and seamlessly modify the underlying metric
for the ML algorithm to LCOT. We show that the proposed metric is rooted in the
Circular Optimal Transport (COT) and can be considered the linearization of the
COT metric with respect to a fixed reference measure. We provide a theoretical
analysis of the proposed metric and derive the computational complexities for
pairwise comparison of circular probability measures. Lastly, through a set of
numerical experiments, we demonstrate the benefits of LCOT in learning
representations of circular measures.