Optimal frame completions

Abstract

Given a finite sequence of vectors F0 in C d we describe the spectral and geometrical structure of optimal frame completions of F0 obtained by appending 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 F0 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 Lidskii’s inequality from matrix theory.