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1. Massively Distributed Graph Distances

   https://doi.org/10.1109/TSIPN.2020.3022003  

   Moharrer, Armin; Gao, Jasmin; Wang, Shikun; Bento, Jose; Ioannidis, Stratis (January 2020, IEEE Transactions on Signal and Information Processing over Networks)

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2. Bisectors and Pinned Distances

   https://doi.org/10.1007/s00454-019-00122-w  

   Lund, Ben; Petridis, Giorgis (October 2020, Discrete & Computational Geometry)
3. Distances on hills look farther than distances on flat ground: Evidence from converging measures

   https://doi.org/10.3758/s13414-017-1305-x  

   Tenhundfeld, Nathan L.; Witt, Jessica K. (May 2017, Attention, Perception, & Psychophysics)
4. Stereographic Spherical Sliced Wasserstein Distances

   Tran, Huy; Bai, Yikun; Kothapalli, Abihith; Shahbazi, Ashkan; Liu, Xinran; Diaz_Martin, Rocio P; Kolouri, Soheil (July 2024, International Conference on Machine Learning)

   Comparing spherical probability distributions is of great interest in various fields, including geology, medical domains, computer vision, and deep representation learning. The utility of optimal transport-based distances, such as the Wasserstein distance, for comparing probability measures has spurred active research in developing computationally efficient variations of these distances for spherical probability measures. This paper introduces a high-speed and highly parallelizable distance for comparing spherical measures using the stereographic projection and the generalized Radon transform, which we refer to as the Stereographic Spherical Sliced Wasserstein (S3W) distance. We carefully address the distance distortion caused by the stereographic projection and provide an extensive theoretical analysis of our proposed metric and its rotationally invariant variation. Finally, we evaluate the performance of the proposed metrics and compare them with recent baselines in terms of both speed and accuracy through a wide range of numerical studies, including gradient flows and self-supervised learning. Our code is available at https://github.com/mint-vu/s3wd. 

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5. Computing distances on Riemann surfaces

   https://doi.org/10.1088/1751-8121/ad653a  

   Stepanyants, Huck; Beardon, Alan; Paton, Jeremy; Krioukov, Dmitri (August 2024, Journal of Physics A: Mathematical and Theoretical)

   Abstract Riemann surfaces are among the simplest and most basic geometric objects. They appear as key players in many branches of physics, mathematics, and other sciences. Despite their widespread significance, how to compute distances between pairs of points on compact Riemann surfaces is surprisingly unknown, unless the surface is a sphere or a torus. This is because on higher-genus surfaces, the distance formula involves an infimum over infinitely many terms, so it cannot be evaluated in practice. Here we derive a computable distance formula for a broad class of Riemann surfaces. The formula reduces the infimum to a minimum over an explicit set consisting of finitely many terms. We also develop a distance computation algorithm, which cannot be expressed as a formula, but which is more computationally efficient on surfaces with high genuses. We illustrate both the formula and the algorithm in application to generalized Bolza surfaces, which are a particular class of highly symmetric compact Riemann surfaces of any genus greater than 1. 

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