Optimal frame completions with prescribed norms for majorization

TL;DR

This paper characterizes and constructs optimal vector completions of a given sequence in complex space, minimizing majorization of eigenvalues of frame operators with prescribed norms, using explicit algorithms and eigenspace analysis.

Contribution

It provides a complete characterization and explicit algorithm for optimal frame completions with prescribed norms based on majorization, extending the Schur-Horn theorem.

Findings

Constructs a vector minimizing majorization among eigenvalues of frame completions.

Describes all optimal completions using eigenspaces of the initial sequence.

Shows that convex potential minimizers are structural and independent of the specific convex function.

Abstract

Given a finite sequence of vectors $\mathcal F_0$ in $\C^d$ we characterize in a complete and explicit way the optimal completions of $\mathcal F_0$ obtained by adding a finite sequence of vectors with prescribed norms, where optimality is measured with respect to majorization (of the eigenvalues of the frame operators of the completed sequence). Indeed, we construct (in terms of a fast algorithm) a vector - that depends on the eigenvalues of the frame operator of the initial sequence $\cF_0$ and the sequence of prescribed norms - that is a minimum for majorization among all eigenvalues of frame operators of completions with prescribed norms. Then, using the eigenspaces of the frame operator of the initial sequence $\cF_0$ we describe the frame operators of all optimal completions for majorization. Hence, the concrete optimal completions with prescribed norms can be obtained using recent algorithmic constructions related with the Schur-Horn theorem. The well known relation between majorization and tracial inequalities with respect to convex functions allow to describe our results in the following equivalent way: given a finite sequence of vectors $\mathcal F_0$ in $\C^d$ we show that the completions with prescribed norms that minimize the convex potential induced by a strictly convex function are structural minimizers, in the sense that they do not depend on the particular choice of the convex potential.