Shift-enabled graphs: Graphs where shift-invariant filters are representable as polynomials of shift operations

TL;DR

This paper introduces the concept of shift-enabled graphs in graph signal processing, highlighting that shift-invariance and polynomial representability depend on specific matrix properties, which impacts filter design.

Contribution

It defines shift-enabled graphs, provides examples of non-shift-enabled graphs, and demonstrates that shift-invariant filters may not always be polynomial functions of the adjacency matrix.

Findings

Not all graphs are shift-enabled, affecting filter representation.

Shift-invariance may not be preserved under graph modifications.

Further research is needed to understand shift-enabled graphs in practice.

Abstract

In digital signal processing, shift-invariant filters can be represented as a polynomial expansion of a shift operation,that is, the Z-transform representation. When extended to graph signal processing (GSP), this would mean that a shift-invariant graph filter can be represented as a polynomial of the adjacency (shift) matrix of the graph. However, the characteristic and minimum polynomials of the adjacency matrix must be identical for the property to hold. While it has been suggested that this condition might be ignored as it is always possible to find a polynomial transform to represent the original adjacency matrix by another adjacency matrix that satisfies the condition, this letter shows that a filter that is shift invariant in terms of the original graph may not be shift invariant anymore under the modified graph and vice versa. We introduce the notion of "shift-enabled graph" for graphs that satisfy the aforementioned condition, and present a concrete example of a graph that is not "shift-enabled" and a shift-invariant filter that is not a polynomial of the shift operation matrix. The result provides a deeper understanding of shift-invariant filters when applied in GSP and shows that further investigation of shift-enabled graphs is needed to make it applicable to practical scenarios.