Frame potentials and the geometry of frames

TL;DR

This paper explores the geometric structure of optimal frames for frame potentials, introducing equidistributed Parseval frames and analyzing the convergence of gradient descent to critical points.

Contribution

It introduces equidistributed Parseval frames, generalizes known classes, and analyzes the geometric and convergence properties of frame potential optimizers.

Findings

Gradient descent on real analytic frame potentials converges to critical points.

Equidistributed Parseval frames can coincide with Grassmannian frames.

Optimal frames are characterized as equal-norm, equipartitioned, or equidistributed.

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 Lojasiewicz, 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.