Time-varying Graph Signals Research Articles

Graph Learning Based on Spatiotemporal Smoothness for Time-Varying Graph Signal

Graph learning often boils down to uncovering the hidden structure of data, which has been applied in various fields such as biology, sociology, and environmental studies. However, distributed sensing in realistic application often gives rise to spatiotemporal signals, which can be characterized through new tools of graph signal processing as a time-varying graph signal. It calls upon the development from static graph signal studies to the joint space-time analysis. In this paper, we study the problem of learning graphs from time-varying graph signals. Based on the correlated properties in observed signals, a dynamic graph-based model is first presented, which particularly takes into account space-time interactions in signal representation. Considering the case that the time correlation pattern is unavailable, the graph learning problem is cast as a joint correlation detecting and graph refining problem. Then it is solved by the proposed correlation-aware and spatiotemporal smoothness-based graph learning method (CASTS), which novelly introduces the spatiotemporal smooth prior to the field of time-vertex signal analysis. By promoting such smoothness in each learning steps, the graph learning accuracy can be efficiently improved. The experiments on both synthetic and real-world datasets demonstrate the improvement of the proposed CASTS over current state-of-the-art graph learning methods, and meanwhile show the capability of dynamic prediction in climate analysis.

Spatio-Temporal Signal Recovery Based on Low Rank and Differential Smoothness

The analysis of spatio-temporal signals plays an important role in various fields including sociology, climatology, and environmental studies, etc. Due to the abrupt breakdown of the sensors in the sensor network, there always are missing entries in the observed spatio-temporal signals. In this paper, we study the problem of recovering spatio-temporal signals from partially known entries. Based on both the global and local correlated property of spatio-temporal signals, we propose a low rank and differential smoothness based recovery method (LRDS), which novelly introduces the differential smooth prior of time-varying graph signals to the field of spatio-temporal signal analysis. The performance of the proposed method is analyzed theoretically. Considering the case where a priori information about the signal's global pattern is available, we propose prior LRDS to further improve the reconstruction accuracy. Such improvement is also verified by synthetic experiments. Besides, experiments on several real-world datasets demonstrate the improvement on recovery accuracy of the proposed LRDS over the state-of-the-art spatio-temporal signal recovery methods.

Forecasting Time Series With VARMA Recursions on Graphs

Graph-based techniques emerged as a choice to deal with the dimensionality issues in modeling multivariate time series. However, there is yet no complete understanding of how the underlying structure could be exploited to ease this task. This paper provides contributions in this direction by considering the forecasting of a process evolving over a graph. We make use of the (approximate) time-vertex stationarity assumption, i.e., time-varying graph signals whose first- and second-order statistical moments are invariant over time and correlated to a known graph topology. The latter is combined with vector autoregressive and vector autoregressive moving average models to tackle the dimensionality issues present in predicting the temporal evolution of multivariate time series. We find out that by projecting the data to the graph spectral domain the multivariate model estimation reduces to that of fitting a number of uncorrelated univariate autoregressive-moving-average models and an optimal low-rank data representation can be exploited so as to further reduce the estimation costs. In the case that the multivariate process can be observed at a subset of nodes, the proposed models extend naturally to Kalman filtering on graphs allowing for optimal tracking. Numerical experiments with both synthetic and real data validate the proposed approach and highlight its benefits over state-of-the-art alternatives.

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.

Time-Varying Graph Signal Reconstruction

Signal processing on graphs is an emerging research field dealing with signals living on an irregular domain that is captured by a graph, and has been applied to sensor networks, machine learning, climate analysis, etc. Existing works on sampling and reconstruction of graph signals mainly studied static bandlimited signals. However, many real-world graph signals are time-varying, and they evolve smoothly, so instead of the signals themselves being bandlimited or smooth on graph, it is more reasonable that their temporal differences are smooth on graph. In this paper, a new batch reconstruction method of time-varying graph signals is proposed by exploiting the smoothness of the temporal difference signals, and the uniqueness of the solution to the corresponding optimization problem is theoretically analyzed. Furthermore, driven by practical applications faced with real-time requirements, huge size of data, lack of computing center, or communication difficulties between two nonneighboring vertices, an online distributed method is proposed by applying local properties of the temporal difference operator and the graph Laplacian matrix. Experiments on a variety of synthetic and real-world datasets demonstrate the excellent performance of the proposed methods.

Autoregressive Moving Average Graph Filtering

One of the cornerstones of the field of signal processing on graphs are graph filters, direct analogues of classical filters, but intended for signals defined on graphs. This work brings forth new insights on the distributed graph filtering problem. We design a family of autoregressive moving average (ARMA) recursions, which (i) are able to approximate any desired graph frequency response, and (ii) give exact solutions for tasks such as graph signal denoising and interpolation. The design philosophy, which allows us to design the ARMA coefficients independently from the underlying graph, renders the ARMA graph filters suitable in static and, particularly, time-varying settings. The latter occur when the graph signal and/or graph are changing over time. We show that in case of a time-varying graph signal our approach extends naturally to a two-dimensional filter, operating concurrently in the graph and regular time domains. We also derive sufficient conditions for filter stability when the graph and signal are time-varying. The analytical and numerical results presented in this paper illustrate that ARMA graph filters are practically appealing for static and time-varying settings, as predicted by theoretical derivations.

A Distributed Tracking Algorithm for Reconstruction of Graph Signals

The rapid development of signal processing on graphs provides a new perspective for processing large-scale data associated with irregular domains. In many practical applications, it is necessary to handle massive data sets through complex networks, in which most nodes have limited computing power. Designing efficient distributed algorithms is critical for this task. This paper focuses on the distributed reconstruction of a time-varying bandlimited graph signal based on observations sampled at a subset of selected nodes. A distributed least square reconstruction (DLSR) algorithm is proposed to recover the unknown signal iteratively, by allowing neighboring nodes to communicate with one another and make fast updates. DLSR uses a decay scheme to annihilate the out-of-band energy occurring in the reconstruction process, which is inevitably caused by the transmission delay in distributed systems. Proof of convergence and error bounds for DLSR are provided in this paper, suggesting that the algorithm is able to track time-varying graph signals and perfectly reconstruct time-invariant signals. The DLSR algorithm is numerically experimented with synthetic data and real-world sensor network data, which verifies its ability in tracking slowly time-varying graph signals.