A Time-Vertex Signal Processing Framework: Scalable Processing and Meaningful Representations for Time-Series on Graphs

An emerging way to deal with high-dimensional noneuclidean data is to assume that the underlying structure can be captured by a graph. Recently, ideas have begun to emerge related to the analysis of time-varying graph signals. This paper aims to elevate the notion of joint harmonic analysis to a full-fledged framework denoted as time-vertex signal processing, that links together the time-domain signal processing techniques with the new tools of graph signal processing. This entails three main contributions: A) We provide a formal motivation for harmonic time-vertex analysis as an analysis tool for the state evolution of simple partial differential equations on graphs; b) we improve the accuracy of joint filtering operators by up-to two orders of magnitude; c) using our joint filters, we construct time-vertex dictionaries analyzing the different scales and the local time-frequency content of a signal. The utility of our tools is illustrated in numerous applications and datasets, such as dynamic mesh denoising and classification, still-video inpainting, and source localization in seismic events. Our results suggest that joint analysis of time-vertex signals can bring benefits to regression and learning.