Optimal completions of a frame

TL;DR

This paper characterizes the structure of optimal frame completions in complex vector spaces using convex optimization, providing explicit algorithms and insights for minimizing potentials like MSE and frame potential.

Contribution

It introduces a method to determine optimal frame completions by reducing the problem to convex optimization and explicitly computing the solutions.

Findings

Optimal completions are characterized by a finite set computable via an algorithm.

The problem reduces to minimizing a convex function over a convex polytope.

The approach applies to potentials like MSE and Benedetto-Fickus' frame potential.

Abstract

Given a finite sequence of vectors $\mathcal F_0$ in $\C^d$ we describe the spectral and geometrical structure of optimal completions of $\mathcal F_0$ obtained by adding a finite sequence of vectors with prescribed norms, where optimality is measured with respect to a general convex potential. In particular, our analysis includes the so-called Mean Square Error (MSE) and the Benedetto-Fickus' frame potential. On a first step, we reduce the problem of finding the optimal completions to the computation of the minimum of a convex function in a convex compact polytope in $\R^d$. As a second step, we show that there exists a finite set (that can be explicitly computed in terms of a finite step algorithm that depends on $\cF_0$ and the sequence of prescribed norms) such that the optimal frame completions with respect to a given convex potential can be described in terms of a distinguished element of this set. As a byproduct we characterize the cases of equality in Lindskii's inequality from matrix theory.