Optimal Dual Frame Pairs: A Synergy with Graph Theory

TL;DR

This paper explores optimizing dual frame pairs for data transmission errors using graph theory, identifying conditions for optimality in erasure scenarios and analyzing error operators.

Contribution

It introduces a graph-theoretic approach to optimize dual frames, including conditions for their optimality in erasure problems and spectral radius computations.

Findings

Tight frames from connected graphs are optimal for one erasure.

Spectral radius of error operators is computed for one and two erasures.

Necessary conditions for dual pair optimality are established.

Abstract

This paper investigates the optimization of dual frame pairs in the context of erasure problems in data transmission, using a graph theoretical approach. Frames are essential for mitigating errors and signal loss due to their redundancy properties. We address the use of spectral radius and operator norm for error measurements, presenting conditions for the optimality of dual pairs for one and two erasures. Our study shows that a tight frame generated by connected graphs and its canonical dual pair is optimal for one-erasure scenarios. Additionally, we compute the spectral radius of the error operator for one and two erasures in graph-generated frames, establishing necessary conditions for dual pair optimality.