Quantile-based fitting for graph signals

Abstract The development of monitoring tools has led to an emerging demand for analyzing data residing on graphs, referred to as graph signals. In this study, we propose a quantile-based fitting method for graph signals, which can be applicable to graph signals with a wide range of distributions. Unlike traditional data fitting methods, such as smoothing splines or quantile smoothing splines in Euclidean space, the proposed method is designed for the graph domain, considering the inherent structure of graphs. In contrast to prevalent graph signal fitting methods that rely on optimization problems with $$L_2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>L</mml:mi> <mml:mn>2</mml:mn> </mml:msub> </mml:math> -norm fidelity, the proposed method provides robust fits for graph signals in the presence of outliers. More importantly, it identifies various distributional structures of graph signals beyond the mean feature. We further investigate the theoretical properties of the proposed solution, including its existence and uniqueness. Through a comprehensive simulation study and real data analysis, we demonstrate the promising performance of the proposed method.