On spectrally optimal duals of frames generated by graphs

TL;DR

This paper investigates spectrally optimal dual frames of graph-generated frames, showing uniqueness for connected graphs and non-uniqueness for disconnected graphs in the context of 1- and 2-erasure scenarios.

Contribution

It characterizes the spectral optimality of dual frames generated by graphs, highlighting the uniqueness for connected graphs and non-uniqueness for disconnected graphs.

Findings

Canonical duals are spectrally optimal for connected graphs.

Non-unique spectrally optimal duals exist for disconnected graphs.

Results apply to scenarios with 1- and 2-erasures.

Abstract

Recently, the concept of frames generated by graphs has been introduced in \cite{D}. In this paper, we study spectrally optimal dual frames of frames generated by graphs. We show that if the frame is generated by a connected graph, then its canonical dual frame is the unique spectrally optimal dual frame for $1$-erasure and $2$-erasures. Further, we show that the canonical dual frames of frames generated by disconnected graphs are non-unique spectrally optimal dual frames for $1$-erasure and $2$-erasures.