Non-orthogonal fusion frames and the sparsity of fusion frame operators

TL;DR

This paper introduces non-orthogonal fusion frames to enhance the sparsity of fusion frame operators, improving distributed system efficiency by analyzing conditions for diagonalization and identity multiples.

Contribution

It proposes the concept of non-orthogonal fusion frames and characterizes when their operators are diagonal or scalar multiples of the identity.

Findings

Fusion frame operators can be made sparse using non-orthogonal frames.

Necessary and sufficient conditions for diagonal and scalar multiple operators are provided.

A scheme for multiple fusion frames with diagonal operators is examined.

Abstract

Fusion frames have become a major tool in the implementation of distributed systems. The effectiveness of fusion frame applications in distributed systems is reflected in the efficiency of the end fusion process. This in turn is reflected in the efficiency of the inversion of the fusion frame operator $S_{\cW}$, which in turn is heavily dependent on the sparsity of $S_{\cW}$. We will show that sparsity of the fusion frame operator naturally exists by introducing a notion of {\it non-orthogonal fusion frames}. We show that for a fusion frame $\{W_i,v_i\}_{i\in I}$, if $\text{dim}(W_i)=k_i$, then the matrix of the non-orthogonal fusion frame operator $\cSw$ has in its corresponding location at most a $k_i\times k_i$ block matrix. We provide necessary and sufficient conditions for which the new fusion frame operator $\cSw$ is diagonal and/or a multiple of an identity. A set of other critical questions are also addressed. A scheme of {\it multiple fusion frames} whose corresponding fusion frame operator becomes an diagonal operator is also examined.