On the paper Optimal dual frames of probabilistic erasures

TL;DR

This paper critically examines previous claims about optimal dual frames for probabilistic erasures, correcting errors and providing a more accurate characterization of 1-erasure probability optimal duals.

Contribution

The authors identify errors in prior proofs, correct the theorem with a stronger condition, and refine the understanding of probability optimal dual frames.

Findings

Counterexamples show previous conditions are insufficient for uniqueness.

A corrected theorem with a stronger condition is proved.

The corollary for tight frames is shown to be incorrect.

Abstract

In the paper Optimal Dual Frames for Probabilistic Erasures, the authors have given conditions under which the canonical dual is claimed to be the unique probability optimal dual for 1-erasure reconstruction. In this paper, we demonstrate via counterexamples that the conditions provided are not sufficient to guarantee uniqueness. We also noticed a mistake in the proof of the theorem and proved the correct version of the theorem with a stronger but valid condition. Furthermore, we show that the corollary asserting uniqueness for a tight frame assumption is also incorrect. Our results refine the understanding of probability optimal dual frame constructions and offer a more complete characterization of the 1-erasure probability optimal duals.