  Polynomial graph filters and their inverses play important roles in graph
signal processing. An advantage of polynomial graph filters is that they can be
implemented in a distributed manner, which involves data transmission between
adjacent vertices only. The challenge arisen in the inverse filtering is that a
direct implementation may suffer from high computational burden, as the inverse
graph filter usually has full bandwidth even if the original filter has small
bandwidth. In this paper, we consider distributed implementation of the inverse
filtering procedure for a polynomial graph filter of multiple shifts, and we
propose two iterative approximation algorithms that can be implemented in a
distributed network, where each vertex is equipped with systems for limited
data storage, computation power and data exchanging facility to its adjacent
vertices. We also demonstrate the effectiveness of the proposed iterative
approximation algorithms to implement the inverse filtering procedure and their
satisfactory performance to denoise time-varying graph signals and a data set
of US hourly temperature at 218 locations.
