Electronic Communications in Probability

Energy optimization for distributions on the sphere and improvement to the Welch bounds

Yan Shuo Tan

Full-text: Open access


For any Borel probability measure on $\mathbb{R} ^n$, we may define a family of eccentricity tensors. This new notion, together with a tensorization trick, allows us to prove an energy minimization property for rotationally invariant probability measures. We use this theory to give a new proof of the Welch bounds, and to improve upon them for collections of real vectors. In addition, we are able to give elementary proofs for two theorems characterizing probability measures optimizing one-parameter families of energy integrals on the sphere. We are also able to explain why a phase transition occurs for optimizers of these two families.

Article information

Electron. Commun. Probab., Volume 22 (2017), paper no. 43, 12 pp.

Received: 23 January 2017
Accepted: 7 July 2017
First available in Project Euclid: 15 August 2017

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60E15: Inequalities; stochastic orderings 52A40: Inequalities and extremum problems 15A69: Multilinear algebra, tensor products

energy minimization frame potentials Welch bounds

Creative Commons Attribution 4.0 International License.


Tan, Yan Shuo. Energy optimization for distributions on the sphere and improvement to the Welch bounds. Electron. Commun. Probab. 22 (2017), paper no. 43, 12 pp. doi:10.1214/17-ECP73. https://projecteuclid.org/euclid.ecp/1502762749

Export citation


  • [1] Noga Alon, Problems and results in extremal combinatorics I, Discrete Mathematics 273 (2003), nos 1–3, 31–53.
  • [2] Patrick Billingsley, Probability and Measure - Third Edition, 1995.
  • [3] Dmitriy Bilyk and Feng Dai, Geodesic distance Riesz energy on the sphere, arXiv:1612.08442v1, (2016).
  • [4] Dmitriy Bilyk, Feng Dai, and Ryan Matzke, Stolarsky principle and energy optimization on the sphere, arXiv:1611.04420v1 (2016).
  • [5] Göran Björck, Distributions of positive mass, which maximize a certain generalized energy integral, Arkiv för matematik 3 (1956), no. 3, 255–269.
  • [6] E Çinlar, Probability and Stochastics, Graduate Texts in Mathematics, vol. 261, Springer, New York, 2011.
  • [7] S. Datta, S. Howard, and D. Cochran, Geometry of the Welch bounds, Linear Algebra and Its Applications 437 (2012), no. 10, 2455–2470.
  • [8] D. L. Donoho and M. Elad, Optimally sparse representation in general (nonorthogonal) dictionaries via $l^1$ minimization, Proceedings of the National Academy of Sciences 100 (2003), no. 5, 2197–2202.
  • [9] M. Ehler and K. A. Okoudjou, Minimization of the probabilistic p-frame potential, Journal of Statistical Planning and Inference 142 (2012), no. 3, 645–659.
  • [10] B. Venkov, Réseaux et designs sphériques, Réseaux Euclidiens, Designs Sphériques et Formes Modulaires, Monogr. Enseign. Math. 37, Enseignement Math., Gèneve, 2001.
  • [11] Roman Vershynin, Introduction to the non-asymptotic analysis of random matrices, Compressed Sensing (Yonina C. Eldar and Gitta Kutyniok, eds.), Cambridge University Press, Cambridge, 2011, pp. 210–268.
  • [12] L. Welch, Lower bounds on the maximum cross correlation of signals, IEEE Transactions on Information Theory 20 (1974), no. 3, 397–399.