Optimal frame completions

P. MASSEY, M. A. RUIZ Y D. STOJANOFF

ADVANCES IN COMPUTATIONAL MATHEMATICS

SPRINGER

Lugar: Berlin; Año: 2014 vol. 40 p. 1011 - 1042

1019-7168

Given a finite sequence of vectors $mathcal F_0$ in $C^d$ we describe the spectral and geometrical structure of optimal frame completions of $mathcal F_0$ 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 $cF_0$ 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 equalityin Lidskii´s inequality from matrix theory.