Minimization of the probabilistic p-frame potential

Abstract

We investigate the optimal configurations of n points on the unit sphere for a class of potential functions. In particular, we characterize these optimal configurations in terms of their approximation properties within frame theory. Furthermore, we consider similar optimal configurations in terms of random distributions of points on the sphere. In this probabilistic setting, we characterize these optimal distributions by means of special classes of probabilistic frames. Our work also indicates some connections between statistical shape analysis and frame theory.

On the Fejes Tóth problem about the sum of angles between lines

“…Conversely, since the minimum in (2.2) is independent of N , the weak * -density of probability measures shows that the result of Benedetto and Fickus also implies (2.1). Moreover, inequality (2.1) for arbitrary measures µ ∈ B(S d ) has been stated in [7]. We have included the nice and short proof above for the sake of completeness.…”

Frame Potentials and Orthogonal Vectors

“…Describing minimizers for the p-frame potential for p not even appears to be a difficult problem, and in general not much is known about the structure of minimizers outside a few exceptional cases (some results can found in the papers of Ehler and Okoudjou in this line [4] or in the recent preprint [2]). A large part of the literature surrounding these energies focuses on their relationship with certain symmetric minimal coherence systems of vectors known as equiangular tight frames (ETFs), studies of which first appeared in the discrete geometry community [8].…”

“…The proof for the above inequality is an extension of the method used in [5], and by restriction to m = 1, one obtains the result proved there. In the limit, it is known that the unique symmetric Borel probability measure which minimizes the pframe energy on the sphere S d−1 equally distributes mass over the vertices of a cross-polytope whenever p ∈ (0, 2) [4]. These energies do not depend on the sign of any vector and so one can reflect any vector about the origin to obtain the same energy.…”

Optimal Properties of the Canonical Tight Probabilistic Frame

“…The notion of probabilistic frames was first introduced in [8] in the setting of probability measures on the unit sphere, and was later generalized to probability measures on R d in [10]. In essence, this theory is a generalization of the theory of finite frames which has seen a wealth of activities in recent year, [6,7,11,12,14].…”