Designing optimal dual frames for $\ell^p-$average error optimization

TL;DR

This paper studies how to select optimal dual frames for signal reconstruction that minimize the average error under various norms, especially in the presence of erasures, providing theoretical characterizations and conditions for optimality.

Contribution

It introduces a new framework for optimal dual frame selection based on error operator norms, extending classical methods and characterizing optimality conditions for multiple erasures.

Findings

Canonical dual frames are uniquely optimal under certain conditions.

Complete characterization of optimal duals for uniform tight frames with multiple erasures.

Identifies relationships between different norm-based optimality criteria.

Abstract

In this paper, we investigates the problem of optimal dual frame selection for signal reconstruction in the presence of erasures. Unlike traditional approaches relying on left inverses, we evaluate performance through the norms of error operators, using the Frobenius norm, spectral radius, and numerical radius as measures. Our central focus is the characterization of dual frames that minimize the $\ell^p-$average under these error operator measurements over all possible erasure patterns. We provide conditions under which the canonical dual frame is uniquely optimal and extend our results to multiple erasures. In the Frobenius norm case, we offer a complete characterization for any number of erasures in uniform tight frames. The paper also examines interconnections between optimality criteria across different norm measures and gives sufficient conditions ensuring uniqueness of the optimal dual.