Optimal Frame Completions with Prescribed Norms for Majorization

Abstract

Given a finite sequence of vectors F 0 in C d we characterize in a complete and explicit way the optimal completions of 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 F 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 F 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 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.

Optimal frame designs for multitasking devices with weight restrictions

“…The first results were obtained for the frame potential in [4] and in a more general context in [9]. The case of general convex potentials was studied in [17,18,21,22,23,24,25,26] (in some cases in the more general setting of frame completion problems with prescribed norms).…”

“…It is then natural to wonder whether there are tight frames with norms prescribed by α. This question has motivated the study of the frame design problem (see [1,8,10,11,14,15,16,20] and [17,18,22,21,24,25,26] for the more general frame completion problem with prescribed norms). It is well known that in some cases there are no tight frames in the class of sequences in C d with norms prescribed by α; in these cases, it is natural to consider minimizers of the frame potential within this class, since the eigenvalues of the frame operator of such minimizers have minimal spread (thus, inducing more stable linear reconstruction processes).…”

Local Lidskii's theorems for unitarily invariant norms

“…In [18] any such F is called a completion of F 0 by a family G, with norms prescribed by the sequence a. Eq. (19) can be used to show items 1. and 2. in Theorem 4.1 for a u.i.n. N .…”

“…In order to get information about local minimizers of Θ (N, S, a) from Eq. (19) we should assume further that N is the Frobenius norm. This obstruction to the general case of item 3.…”

Frame Theory and Optimal Anchor Geometries in Wireless Localization

“…Another example is the truncated icosahedron, also known as the soccerbal [8]. Whenever the noise models for the anchors are not identical, tight frames can still be obtained such as described in [9], where a frame construction procedure, similar to the GramSchmidt method for orthogonal bases is proposed.…”