Optimal dual frames and frame completions for majorization

TL;DR

This paper characterizes optimal frame completions and dual frames in terms of majorization and submajorization, providing explicit structures and solutions for minimizing convex functionals and Frobenius norms.

Contribution

It introduces explicit spectral and geometric descriptions of optimal frame completions and dual frames under majorization constraints, advancing frame theory methods.

Findings

Optimal completions minimize convex functionals including mean square error.

Explicit structure of dual frames with bounded Frobenius norms is provided.

Spectral and geometric characterizations are derived for matrices minimizing submajorization.

Abstract

In this paper we consider two problems in frame theory. On the one hand, given a set of vectors $\mathcal F$ we describe the spectral and geometrical structure of optimal completions of $\mathcal F$ by a finite family of vectors with prescribed norms, where optimality is measured with respect to majorization. In particular, these optimal completions are the minimizers of a family of convex functionals that include the mean square error and the Bendetto-Fickus' frame potential. On the other hand, given a fixed frame $\mathcal F$ we describe explicitly the spectral and geometrical structure of optimal frames $\mathcal G$ that are in duality with $\mathcal F$ and such that the Frobenius norms of their analysis operators is bounded from below by a fixed constant. In this case, optimality is measured with respect to submajorization of the frames operators. Our approach relies on the description of the spectral and geometrical structure of matrices that minimize submajorization on sets that are naturally associated with the problems above.