Discrete Energy on Rectifiable Sets

By S. Borodachov, D. Hardin, Edward B. Saff. Springer Monographs in Mathematics, 2019

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Dissecting Supervised Constrastive Learning (2021, methodology): “Even for the related and more studied Thomson problem of minimizing the potential energy of K equally charged particles on the 2-dimensional sphere, the minimizers are only known for K ∈ { 2 , 3 , 4 , 5 , 6 , 12 } (Borodachov et al., 2019).”
Understanding Contrastive Representation Learning through Alignment and Uniformity on the Hypersphere (2020, methodology): “To this end, we consider the Gaussian potential kernel (also known as the Radial Basis Function (RBF) kernel) Gt : S d × Sd → R+ (Cohn & Kumar, 2007; Borodachov et al., 2019): Gt(u, v) , e −t‖u−v‖2 2 = e2t·u Tv−2t, t > 0, and define the uniformity loss as the logarithm of the average pairwise…”