MINIMUM-ENERGY MEASURES FOR SINGULAR KERNELS

Luc Pronzato (1, 2) , Anatoly A. Zhigljavsky (3)

1 Laboratoire I3S - SIS - Signal, Images et Systèmes
2 GdR MASCOT-NUM - Méthodes d'Analyse Stochastique des Codes et Traitements Numériques
3 School of Mathematics [Cardiff]

We develop algorithms for energy minimization for kernels with singularities. This problem arises in different fields, most notably in construction of space-filling sequences of points where singularity of kernels guarantees a strong repelling property between these points. Numerical algorithms are based on approximating singular kernels by non-singular ones, subsequent discretisation and solving non-singular discrete problems. For approximating singular kernels, we approximate an underlying completely monotone (briefly, CM) function with singularity by a bounded CM function with controlled accuracy. Theoretical properties of the suggested approximation are studied and some numerical results are shown.

Mots clés: energy minimization; singular kernels; Riesz kernel; Riesz potential; space-filling design; AMS subject classification: 62K99; 65D30; 65D99

Domaines: Statistiques [math.ST]

Luc Pronzato, Anatoly A. Zhigljavsky. MINIMUM-ENERGY MEASURES FOR SINGULAR KERNELS. 2020. ⟨hal-02495643v1⟩