Quantization Analysis and Robust Design for Distributed Graph Filters

Abstract

Distributed graph filters have recently found applications in wireless sensor networks (WSNs) to solve distributed tasks such as reaching consensus, signal denoising, and reconstruction. However, when implemented over WSNs, the graph filters should deal with network limited energy constraints as well as processing and communication capabilities. Quantization plays a fundamental role to improve the latter but its effects on distributed graph filtering are little understood. WSNs are also prone to random link losses due to noise and interference. In this instance, the filter output is affected by both the quantization error and the topological randomness error, which, if it is not properly accounted in the filter design phase, may lead to an accumulated error through the filtering iterations and significantly degrade the performance. In this paper, we analyze how quantization affects distributed graph filtering over both time-invariant and time-varying graphs. We bring insights on the quantization effects for the two most common graph filters: the finite impulse response (FIR) and autoregressive moving average (ARMA) graph filter. Besides providing a comprehensive analysis, we devise theoretical performance guarantees on the filter performance when the quantization stepsize is fixed or changes dynamically over the filtering iterations. For FIR filters, we show that a dynamic quantization stepsize leads to more reduction of the quantization noise than in the fixed-stepsize quantization. For ARMA graph filters, we show that decreasing the quantization stepsize over the iterations reduces the quantization noise to zero at the steady-state. In addition, we propose robust filter design strategies that minimize the quantization noise for both timeinvariant and time-varying networks. Numerical experiments on synthetic and two real data sets corroborate our findings and show the different trade-offs between quantization bits, filter order, and robustness to topological randomness.

Computation of Graph Fourier Transform Centrality Using Graph Filter

“…5 can be realized by using node-to-node communication and local node processor in the graph network. Recently, the quantization analysis and robust design for distributed graph filters have been investigated in [25], but the circuit hardware implementation is still not introduced. Therefore, it is an interesting research topic to study the circuit implementation of distributed graph filters in the future.…”

A Cascaded Structure for Generalized Graph Filters

“…Furthermore, if traditional time-domain designs are extended to the graph setting, the obtained filters will be restricted to be shift invariant, i.e., polynomials of the GSO, leading to inappropriate filters for approximating arbitrary linear operators by means of graph filters. Moreover, the works on robust design for GFs mostly focus on classical GFs and model-driven settings or on the theoretical understanding of effects such as quantization in the GF processing chain, see, e.g., [12], [13], [40]. Hence they either do not address the data-driven setting for (classical) generalized GFs case and do not deal with the numerically problems in the design stage due to the conditioning of the matrices.…”