Distributed Nonlinear Polynomial Graph Filter and Its Output Graph Spectrum: Filter Analysis and Design

Abstract

While frequency-domain algorithms have been demonstrated to be powerful for conventional nonlinear signal processing, there is still not much progress in literature dedicated to nonlinear graph signal processing in frequency domain. The difficulties come from the irregular structure of graph signals and the lack of a straightforward description for the nonlinear relationship between input and output graph signals. In this paper, a distributed nonlinear polynomial graph filter (NPGF) is proposed to characterize the nonlinear input-output relationship by employing polynomial-type nonlinearity based on Hadamard product. Then, the nonlinear output graph signal is reformulated based on the multi-dimensional inverse graph Fourier transform (GFT), where the m-dimensional GFT coefficients are defined as explicit functions of input graph spectrum and filter structural factors (filter parameters, graph shift matrix, nonlinear degree, and graph shift order). The effects of filter structural factors on the output graph spectrum are further evaluated by proposing graph frequency response (GFR) coefficients, which inherently characterize the irregular structural behaviors of a distributed NPGF in frequency domain. The nonlinear output graph spectrum is then developed based on GFR vectors by mapping the GFR coefficients in the tensor formulation to the output GFT coefficients in the vector expression. This can greatly improve the efficiency in calculating the output graph signal if the input signal only involves a few frequency components. Finally, algorithms are proposed to design the filter parameters for the desired output signal. Numerical studies are presented to illustrate and validate the proposed frequency-domain framework for nonlinear graph signal processing.