Minimal Riesz Energy Point Configurations for Rectifiable d-Dimensional Manifolds

By D. Hardin, Edward B. Saff. 2003

167 citations recorded by the source corpus.

We investigate the energy of arrangements of N points on a rectifiable d-dimensional manifold A R d 0 that interact through the power law (Riesz) potential V = 1/r s , where s > 0 and r is Euclidean distance in R d 0 . With Es(A,N) denoting the minimal energy for such N-point configurations, we determine the asymptotic behavior (as N ! 1) of Es(A,N) for each fixed s d. Moreover, if A has positive d-dimensional Hausdor measure, we show that N-point configurations on A that minimize the s-energy are asymptotically uniformly distributed with respect to d-dimensional Hausdor measure on A when s d. Even for the unit sphere S d R d+1 , these results are new.

Subfields: Materials Chemistry, Numerical Analysis, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/6776d977898bc094f31f42b9d2d8848d390d4d05

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In Materials Chemistry

1. Bo Dai
2. Le Song
3. Mathieu Lewin
4. Dario L. Ringach
5. Xavier Blanc

In Numerical Analysis

1. Bo Dai
2. Le Song
3. Phillip Isola
4. Mathieu Lewin
5. Dario L. Ringach
6. Tongzhou Wang
7. Z. Liu
8. Xavier Blanc

In Statistics and Probability

1. Bo Dai
2. Rongmei Lin
3. James M. Rehg
4. Le Song
5. Li Xiong
6. Phillip Isola
7. Zhen Liu
8. Mathieu Lewin

Citations in the corpus

• Orthogonal Over-Parameterized Training (2020, background): “From [19], we also have the following lemma that shows the same conclusion holds for the the case of s = d and s > d : Lemma 3.”
• The Crystallization Conjecture: A Review (2015, background): “If s > d , it was proved in [32, 120, 126, 127, 143, 172] that the corresponding term reads”
• Understanding Contrastive Representation Learning through Alignment and Uniformity on the Hypersphere (2020, background): “…a certain kernel function (Borodachov et al., 2019; Landkof, 1972), e.g., the Thomson problem of finding the minimal electrostatic potential energy configuration of electrons (Thomson, 1904), and minimization of the Riesz s-potential (Götz & Saff, 2001; Hardin & Saff, 2005; Liu et al., 2018).”
• Learning towards Minimum Hyperspherical Energy (2018, background): “This section leverages a number of rigorous theoretical results from [38, 23, 12, 25, 11, 23, 8, 54] and provides theoretical yet intuitive understandings about MHE.”
• Population coding under normalization (2009, background): “This problem is a generalization of the classic Thompson problem of the distribution of charges on a sphere (Thomson, 1904) and other manifolds (Borodachov, Hardin, & Saff, 2007, 2008; Hardin & Saff, 2005).”