On semi shift invariant graph filters

TL;DR

This paper introduces semi shift invariant graph filters, generalizing traditional shift invariant filters to overcome their limitations, and demonstrates their applications in subgraph signal processing and graph neural networks.

Contribution

The paper proposes semi shift invariant filters, expanding the class of graph filters and enabling more flexible applications in graph signal processing and neural networks.

Findings

Semi shift invariant filters generalize shift invariant filters.

Application of semi shift invariant filters in subgraph signal processing.

Use of semi shift invariant filters in graph neural networks.

Abstract

In graph signal processing, one of the most important subjects is the study of filters, i.e., linear transformations that capture relations between graph signals. One of the most important families of filters is the space of shift invariant filters, defined as transformations commute with a preferred graph shift operator. Shift invariant filters have a wide range of applications in graph signal processing and graph neural networks. A shift invariant filter can be interpreted geometrically as an information aggregation procedure (from local neighborhood), and can be computed easily using matrix multiplication. However, there are still drawbacks to using solely shift invariant filters in applications, such as being restrictively homogeneous. In this paper, we generalize shift invariant filters by introducing and studying semi shift invariant filters. We give an application of semi shift invariant filters with a new signal processing framework, the subgraph signal processing. Moreover, we also demonstrate how semi shift invariant filters can be used in graph neural networks.