LCOT: Linear circular optimal transport
Rocio Diaz Martin, Ivan Medri, Yikun Bai, Xinran Liu, Kangbai Yan,, Gustavo K. Rohde, Soheil Kolouri
TL;DR
This paper introduces LCOT, a new efficient metric for circular probability measures that enables linear embedding and improves representation learning on non-Euclidean circular data.
Contribution
The paper proposes LCOT, a linearized and computationally efficient metric for circular measures, rooted in Circular Optimal Transport, facilitating ML applications.
Findings
LCOT provides an explicit linear embedding for circular measures.
The metric is computationally efficient for pairwise comparisons.
Numerical experiments show LCOT improves representation learning.
Abstract
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…
Peer Reviews
Decision·ICLR 2024 poster
1. In Fig. 4, this work demonstrated that the LCOT can be computed much faster than the original optimal transport on the circle, which is consistent with the theoretical analysis. 2. The relationship between the original optimal transport on the circle and the LCOT is discussed in Remark 3.4.
1. The intuitive interpretation of LCOT, including Eq. (17) and equations in Def. 3.2, is not clear to the reviewer. 2. The authors claimed that "invertible nature of the LCOT embedding allows us to directly calculate the barycenter" in Sec. 4, but the algorithm is missing. It is important for future research to show detailed algorithms in this paper. 3. In Fig. 4, the unit (e.g., seconds, mili seconds) is missing.
- Paper easy to follow and well written. - Proposing a novel linearization of the circular OT metric (LCOT) by embedding the origin measures via a Monge displacement with respect to reference measures. - LCOT outperforms classical COT in terms of computational complexity. - Extensive numerical experiments comparing COT and LCOT (with different reference measures) on simulated datasets. - The proofs sound good to me.
- There is a backbone result in the paper which is the uniqueness of $\alpha_{\mu, \nu}$ (see Remark 2.4). There is an empirical showing of this uniqueness in Figure 8, Appendix A.2. However, I am still not convinced, since in Equation (8) one can get different cutting points, and using Equation (9), this may give different $\alpha_{\mu, \nu}$. Probably, the result is trivial but I think it deserves to be proved. - As it can be seen in Figure 4, the LCOT with non-uniform reference measure is slo
Overall, the paper is nicely written and very clear. It proposes a new metric and show its benefit compared to the previous Circular Optimal Transport distances. - Very well written and clear - New efficient metric on the circle with a theoretical analysis - Experiments which show the behavior of LCOT in comparison with COT: runtimes, comparison of the distances through a MDS and barycenter computation which cannot be directly done with COT.
- The paper is pretty incremental as it feels like a specification of the Linear OT framework in the particular case of the circle, which relies heavily on previous works such as [1, 2] for the Linear OT framework and [3] for the closed-form on the circle. - The experiments are only on toy data. [1] Sarrazin, Clément, and Bernhard Schmitzer. "Linearized optimal transport on manifolds." arXiv preprint arXiv:2303.13901 (2023). [2] Wang, Wei, Dejan Slepčev, Saurav Basu, John A. Ozolek, and Gustav
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