Frame Potentials and the Geometry of Frames

Abstract

This paper concerns the geometric structure of optimizers for frame potentials. We consider finite, real or complex frames and rotation or unitarily invariant potentials, and mostly specialize to Parseval frames, meaning the frame potential to be optimized is a function on the manifold of Gram matrices belonging to finite Parseval frames. Next to the known classes of equal-norm and equiangular Parseval frames, we introduce equidistributed Parseval frames, which are more general than the equiangular type but have more structure than equal-norm ones. We also provide examples where this class coincides with that of Grassmannian frames, the minimizers for the maximal magnitude among inner products between frame vectors. These different types of frames are characterized in relation to the optimization of frame potentials. Based on results by Łojasiewicz, we show that the gradient descent for a real analytic frame potential on the manifold of Gram matrices belonging to Parseval frames always converges to a critical point. We then derive geometric structures associated with the critical points of different choices of frame potentials. The optimal frames for families of such potentials are thus shown to be equal-norm, or additionally equipartitioned, or even equidistributed.

Achieving the orthoplex bound and constructing weighted complex projective 2-designs with Singer sets

“…This paper is dedicated to the construction of other types of OGFs. By examining the behavior of the gradient descent for a frame potential that is a smooth perturbation of µ(F) [5], we discovered the following OGF in Ω 5,2 .…”

On the Search for Tight Frames of Low Coherence

“…which has been studied in several works including Welch [67], Conway et al [31], Strohmer and Heath [62], and more recently [40,9]. The problem (1.3) is often referred as the best line-packing problem because it asks how to arrange N lines in H d so that they are as far apart as possible.…”

Frame potential for finite-dimensional Banach spaces

“…Significant interest in FUNTFs started in 2003 when Benedetto and Fickus proved that for every k ≥ n, every n-dimensional Hilbert space has a FUNTF of k vectors [BF03]. To do this they defined a positive function on the set {(x j ) k j=1 : x j ∈ H, x j = 1}, called the frame potential, so that a collection of vectors minimizes the frame potential if and only if the vectors form a tight frame for H. Since then, researchers have been interested in understanding the geometry and topology of FUNTFs [BH15] [CMS17] and properties of the frame potential itself [FJKO05] [JO08]. In [DFKLOW04], an algorithm is given to create FUNTFs and in [CF09] it is shown that the method of gradient descent on the frame potential can be used to create FUNTFs.…”