Nonlinear Polynomial Graph Filter for Signal Processing With Irregular Structures

Processing of signals with irregular structures is a fundamental challenge to the conventional signal processing due to the complex structures. Since most of the relationships among such data from engineering applications are inherently nonlinear, those tools that do not take the nonlinearity into account may have limitations in dealing with such relationships. In this paper, a novel nonlinear polynomial graph filter (NPGF) is proposed to facilitate the nonlinear processing of signals with irregular structures. First of all, the nonlinear relationship between the input and output graph signals is constructed, which is further developed to utilize the structure information of the irregular structures (i.e., the graph shift S). Specifically, the nonlinear kernels and the nonlinear input terms are both designed to be polynomials of the graph shift S. Such nonlinear polynomial relationship is then defined as the NPGF. Note that shift invariance is a fundamental property in signal processing but is not straightforward in the existing nonlinear graph signal processing techniques. Due to the special expression of the NPGF, the shift-invariance is extended to our NPGF, which can conveniently facilitate the analysis and design of graph signals using NPGF and is one of the most important contributions in this paper. Numerical studies that characterize the nonlinear temperature distribution and GDP prediction based on real datasets are presented. Compared with other existing techniques, our NPGF achieves superior capability in characterizing the nonlinear relationship between the input and output graph signals, especially when only very few graph signals can be observed.