Three proofs of the Benedetto-Fickus theorem

TL;DR

This paper presents three modern proofs of the Benedetto-Fickus theorem, which characterizes the global minimizers of the frame potential as unit norm tight frames, highlighting the landscape's favorable properties for optimization.

Contribution

It introduces three new proofs of the Benedetto-Fickus theorem using contemporary optimization landscape analysis techniques.

Findings

The frame potential has no spurious local minimizers.

Global minimizers are exactly the unit norm tight frames.

Optimization methods can effectively find these frames.

Abstract

In 2003, Benedetto and Fickus introduced a vivid intuition for an objective function called the frame potential, whose global minimizers are fundamental objects known today as unit norm tight frames. Their main result was that the frame potential exhibits no spurious local minimizers, suggesting local optimization as an approach to construct these objects. Local optimization has since become the workhorse of cutting-edge signal processing and machine learning, and accordingly, the community has identified a variety of techniques to study optimization landscapes. This chapter applies some of these techniques to obtain three modern proofs of the Benedetto-Fickus theorem.