Fast Decentralized Linear Functions Over Edge Fluctuating Graphs

TL;DR

This paper introduces a fast, decentralized method for computing a broad class of linear transformations over dynamic graphs, addressing edge fluctuations and improving efficiency in distributed network signal processing.

Contribution

It develops a novel framework for decentralized linear transformation computation using successive graph shift operators and an online kernel-based method to handle edge fluctuations.

Findings

The proposed method computes transformations in fewer iterations.

It effectively mitigates the impact of edge fluctuations.

The approach reduces communication and power consumption in sensor networks.

Abstract

Implementing linear transformations is a key task in the decentralized signal processing framework, which performs learning tasks on data sets distributed over multi-node networks. That kind of network can be represented by a graph. Recently, some decentralized methods have been proposed to compute linear transformations by leveraging the notion of graph shift operator, which captures the local structure of the graph. However, existing approaches have some drawbacks such as considering some special instances of linear transformations, or reducing the family of transformations by assuming that a shift matrix is given such that a subset of its eigenvectors spans the subspace of interest. In contrast, this paper develops a framework for computing a wide class of linear transformations in a decentralized fashion by relying on the notion of graph shift operator. The main goal of the proposed method is to compute the desired linear transformation in a small number of iterations. To this end, a set of successive graph shift operators is employed, then, a new optimization problem is proposed whose goal is to compute the desired transformation as fast as possible. In addition, usually, the topology of the networks, especially the wireless sensor networks, change randomly because of node failures or random links. In this paper, the effect of edge fluctuations on the performance of the proposed method is studied. To deal with the negative effect of edge fluctuations, an online kernel-based method is proposed which enables nodes to estimate the missed values with their at hand information. The proposed method can also be employed to sparsify the network graph or reduce the number of local exchanges between nodes, which saves sensors power in the wireless sensor networks.