A Class of Doubly Stochastic Shift Operators for Random Graph Signals and their Boundedness

TL;DR

This paper introduces a new class of doubly stochastic graph shift operators that are mathematically bounded and preserve signal properties, with practical applications demonstrated in multi-sensor signal filtering.

Contribution

The paper proposes a novel class of doubly stochastic GSOs with proven boundedness and preservation properties, bridging theoretical analysis and practical filtering applications.

Findings

Proven lower and upper L2-boundedness for locally stationary signals

Demonstrated L2-isometry for i.i.d. signals with increasing neighborhood size

Showed preservation of the mean of any graph signal

Abstract

A class of doubly stochastic graph shift operators (GSO) is proposed, which is shown to exhibit: (i) lower and upper $L_{2}$-boundedness for locally stationary random graph signals; (ii) $L_{2}$-isometry for \textit{i.i.d.} random graph signals with the asymptotic increase in the incoming neighbourhood size of vertices; and (iii) preservation of the mean of any graph signal. These properties are obtained through a statistical consistency analysis of the graph shift, and by exploiting the dual role of the doubly stochastic GSO as a Markov (diffusion) matrix and as an unbiased expectation operator. Practical utility of the class of doubly stochastic GSOs is demonstrated in a real-world multi-sensor signal filtering setting.