Polynomial graph filters of multiple shifts and distributed implementation of inverse filtering

Polynomial graph filters and their inverses play important roles in graph signal processing. In this paper, we introduce the concept of multiple commutative graph shifts and polynomial graph filters, which could play similar roles in graph signal processing as the one-order delay and finite impulse response filters in classical multi-dimensional signal processing. We implement the filtering procedure associated with a polynomial graph filter of multiple shifts at the vertex level in a distributed network on which each vertex is equipped with a data processing subsystem for limited computation power and data storage, and a communication subsystem for direct data exchange only to its adjacent vertices. In this paper, we also consider the implementation of inverse filtering procedure associated with a polynomial graph filter of multiple shifts, and we propose two iterative approximation algorithms applicable in a distributed network and in a central facility. We also demonstrate the effectiveness of the proposed algorithms to implement the inverse filtering procedure on denoising time-varying graph signals and a dataset of US hourly temperature at 218 locations. © 2022, The Author(s), under exclusive licence to Springer Nature Switzerland AG.