Probability Modelled Averaged Spectrally Optimal Dual Frame and Dual Pair for Erasure

TL;DR

This paper introduces Probabilistic Averaged Spectrally Optimal Dual frames (PASOD-frames) for minimizing reconstruction error under erasures, analyzing their properties, uniqueness, and characterizations within frame theory.

Contribution

It defines PASOD-frames and PASOD-pairs based on a new spectral radius and operator norm measurement, and studies their properties, convexity, and conditions for optimality and uniqueness.

Findings

Set of PASOD-frames is convex, closed, and compact.

Image of PASOD-frames under unitary operators remains PASOD.

Conditions for canonical dual to be unique PASOD-frame are established.

Abstract

Finding the optimal dual frame and optimal dual pair for signal reconstruction, which can minimize the reconstruction error when erasure occurs during data transmission, is a deep rooted problem from the perspective of frame theory. In this paper, we consider a new measurement for the error operator by taking the average of spectral radius and operator norm with probabilistic erasure. In this measurement, optimal dual frames are called Probabilistic Averaged Spectrally Optimal Dual frames, PASOD-frames in short and optimal dual pair is called PASOD-pair. The properties of the set of PASOD-frames for a pre-selected frame, has been studied. We prove that the set of all PASOD-frames is convex, closed and compact. We also show that the image of a PASOD-frame and PASOD-pair under any unitary operator is also a PASOD-frame and PASOD-pair. We provide several equivalent conditions for the canonical dual to be the unique PASOD-frame for a given frame $F.$ Moreover, we prove non-uniqueness of PASOD-frame under certain condition of the given frame. We also go on to characterize the set of all PASOD-pairs and give several equivalent conditions for a dual pair to become POD, PSOD and PASOD-pair.