Frame completions with prescribed norms: local minimizers and applications

TL;DR

This paper studies the problem of completing a set of vectors with prescribed norms to minimize convex potentials, showing local minimizers are global and connecting to frame operator distance problems, thus generalizing known results.

Contribution

It proves local minimizers of convex potentials are also global in frame completion problems with prescribed norms, extending results for the frame potential and addressing a generalized Strawn's conjecture.

Findings

Local minimizers are also global minimizers for convex potentials.

Established a connection between frame completion and frame operator distance problems.

Settled a generalized version of Strawn's conjecture on frame operator distance.

Abstract

Let $\mathcal F_0=\{f_i\}_{i\in\mathbb{I}_{n_0}}$ be a finite sequence of vectors in $\mathbb C^d$ and let $\mathbf{a}=(a_i)_{i\in\mathbb{I}_k}$ be a finite sequence of positive numbers. We consider the completions of $\cal F_0$ of the form $\cal F=(\cal F_0,\cal G)$ obtained by appending a sequence $\cal G=\{g_i\}_{i\in\mathbb{I}_k}$ of vectors in $\mathbb C^d$ such that $\|g_i\|^2=a_i$ for $i\in\mathbb{I}_k$, and endow the set of completions with the metric $d(\cal F,\tilde {\mathcal F}) =\max\{ \,\|g_i-\tilde g_i\|: \ i\in\mathbb{I}_k\}$ where $\tilde {\cal F}=(\cal F_0,\,\tilde {\cal G})$. In this context we show that local minimizers on the set of completions of a convex potential $\text{P}_\varphi$, induced by a strictly convex function $\varphi$, are also global minimizers. In case that $\varphi(x)=x^2$ then $\text{P}_\varphi$ is the so-called frame potential introduced by Benedetto and Fickus, and our work generalizes several well known results for this potential. We show that there is an intimate connection between frame completion problems with prescribed norms and frame operator distance (FOD) problems. We use this connection and our results to settle in the affirmative a generalized version of Strawn's conjecture on the FOD.