LCOT: Linear circular optimal transport
This work addresses a domain-specific problem for researchers and practitioners in representation learning dealing with circular data, but it is incremental as it builds on existing Circular Optimal Transport (COT) methods.
The authors tackled the problem of comparing circular probability measures by introducing Linear Circular Optimal Transport (LCOT), a computationally efficient metric with an explicit linear embedding that enables machine learning algorithms to use it, and demonstrated its benefits in representation learning through numerical experiments.
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.