A generalized Schur-Horn theorem and optimal frame completions

TL;DR

This paper generalizes the Schur-Horn theorem to characterize spectra of all possible finite frame completions with given vector lengths and develops an efficient algorithm for optimal frame completion.

Contribution

It introduces a new generalization of the Schur-Horn theorem for finite frame completions and provides a simple algorithm for optimal completion.

Findings

Characterization of spectra for all finite frame completions

A new simple algorithm for optimal frame completion

Extension of classical matrix analysis results to frame theory

Abstract

The Schur-Horn theorem is a classical result in matrix analysis which characterizes the existence of positive semidefinite matrices with a given diagonal and spectrum. In recent years, this theorem has been used to characterize the existence of finite frames whose elements have given lengths and whose frame operator has a given spectrum. We provide a new generalization of the Schur-Horn theorem which characterizes the spectra of all possible finite frame completions. That is, we characterize the spectra of the frame operators of the finite frames obtained by adding new vectors of given lengths to an existing frame. We then exploit this characterization to give a new and simple algorithm for computing the optimal such completion.