Optimal frame completions

PEDRO MASSEY; MARIANO RUIZ; DEMETRIO STOJANOFF

ADVANCES IN COMPUTATIONAL MATHEMATICS

SPRINGER

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

1019-7168

Given a nite sequence of vectors F_0 in C^d we describe the spectral and geometrical structure of optimal frame completions of F_0 obtained by appending a nite 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 rst step, we reduce the problem of nding 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 nite set (that can be explicitly computed in terms of a nite step algorithm that depends on F_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 Lidskii's inequality from matrix theory.