Multiplicative Lidskii's inequalities and optimal perturbations of frames

TL;DR

This paper develops a mathematical framework for optimizing dual frames and frame perturbations in finite-dimensional spaces using multiplicative inequalities and eigenvalue majorization, with applications to minimizing convex potentials.

Contribution

It introduces a multiplicative version of Lidskii's inequality and characterizes optimal perturbations of frames based on eigenvalue submajorization, advancing frame design theory.

Findings

Derived multiplicative Lidskii's inequality for eigenvalues.

Characterized optimal dual frames under norm restrictions.

Identified conditions for equality cases in eigenvalue inequalities.

Abstract

In this paper we study two design problems in frame theory: on the one hand, given a fixed finite frame $\cF$ for $\hil\cong\C^d$ we compute those dual frames $\cG$ of $\cF$ that are optimal perturbations of the canonical dual frame for $\cF$ under certain restrictions on the norms of the elements of $\cG$. On the other hand, for a fixed finite frame $\cF=\{f_j\}_{j\in\In}$ for $\hil$ we compute those invertible operators $V$ such that $V^*V$ is a perturbation of the identity and such that the frame $V\cdot \cF=\{V\,f_j\}_{j\in\In}$ - which is equivalent to $\cF$ - is optimal among such perturbations of $\cF$. In both cases, optimality is measured with respect to submajorization of the eigenvalues of the frame operators. Hence, our optimal designs are minimizers of a family of convex potentials that include the frame potential and the mean squared error. The key tool for these results is a multiplicative analogue of Lidskii's inequality in terms of log-majorization and a characterization of the case of equality.