Probabilistic Modelled Optimal Frame for Erasures under Spectral and Operator Norm

TL;DR

This paper characterizes the optimal dual frames in frame theory that minimize reconstruction error under spectral and operator norms, providing conditions and characterizations for optimality.

Contribution

It introduces new conditions for when the canonical dual is optimal and characterizes the set of dual pairs achieving optimal reconstruction error.

Findings

Canonical dual is optimal under specific conditions.

Set of dual pairs attaining optimality is characterized.

Provides equivalent conditions for optimal dual frames.

Abstract

Error occurs in data transmission process when some data are missing at the time of reconstruction. Finding the best dual frame or a dual pair that minimizes the reconstruction error when erasure occurs,is a deep-rooted problem in frame theory. The main motivation behind this paper is to characterize the optimal dual under the spectral and operator norms. Here we give several equivalent conditions for which the canonical dual is an optimal dual frame for a given frame. We also go on to characterize the set of dual pairs which attains the optimal value.