On spectrally optimal duals for r-erasures of frames generated by graphs

TL;DR

This paper extends the study of spectrally optimal dual frames to r-erasures in graph-generated frames, establishing uniqueness for connected graphs and non-uniqueness for disconnected graphs.

Contribution

It generalizes previous work by analyzing r-erasures, proving spectral radius invariance under unitary transformations, and characterizing the uniqueness of spectrally optimal duals based on graph connectivity.

Findings

Spectral radius of error operators is invariant under unitary equivalence.

Canonical dual frames are uniquely spectrally optimal for connected graph-generated frames.

Disconnected graph-generated frames have multiple spectrally optimal duals.

Abstract

In [9], authors studied spectrally optimal dual frames for 1-erasure and 2-erasures of frames generated by graph. In this paper, we study spectrally optimal dual frames for r-erasures. We show that the spectral radius of the error operator of unitary equivalent frames is same with respect to their respective canonical dual frames. We prove that if a frame is generated by a connected graph, then its canonical dual frame is the unique spectrally optimal dual frame for r-erasures. Further, we show that the canonical dual of frames generated by disconnected graphs are non-unique spectrally optimal dual frames for r-erasures.