  Graph filters and their inverses have been widely used in denoising,
smoothing, sampling, interpolating and learning. Implementation of an inverse
filtering procedure on spatially distributed networks (SDNs) is a remarkable
challenge, as each agent on an SDN is equipped with a data processing subsystem
with limited capacity and a communication subsystem with confined range due to
engineering limitations. In this letter, we introduce a preconditioned gradient
descent algorithm to implement the inverse filtering procedure associated with
a graph filter having small geodesic-width. The proposed algorithm converges
exponentially, and it can be implemented at vertex level and applied to
time-varying inverse filtering on SDNs.
