Stationary time-vertex signal processing

Abstract

The goal of this paper is to improve learning for multivariate processes whose structure is dependent on some known graph topology; especially when the number of available samples is much smaller than the number of variables. Typically, the graph information is incorporated into the learning process via a smoothness assumption postulating that the values supported on well-connected vertices exhibit small variations. We argue that smoothness is not enough. To capture the behavior of complex interconnected systems, such as transportation and biological networks, it is important to train expressive models, being able to reproduce a wide range of graph and temporal behaviors.Motivated by this need, this paper puts forth a novel definition of time-vertex wide-sense stationarity, or joint stationarity for short. We believe that the proposed definition is natural, at it intimately relates to existing definitions of stationarity in the time and vertex domains. We use joint stationarity to regularize learning and to reduce computational complexity in both estimation and recovery tasks. In particular, we show that for any jointly stationary process: (a) one can learn the covariance structure from O(1) samples, and (b) can solve MMSE recovery problems, such as interpolation, denoising, forecasting, in complexity that is linear to the edges and timesteps. Experiments with three datasets suggest that joint stationarity can yield significant accuracy improvements in the reconstruction effort of under-sampled problems, even when the graph is only approximately known or the process is only close to stationary.Proof of Theorem 1. By construction of the JFT basis, X[0, 0] captures the DC-offset of a signal, and condition (a) is equivalent to stating that E[x] = c1 N T . Moreover, if the graph is connected and (a) holds, at least one of E X [n 1 , τ 1 ]and E X [n 2 , τ 2 ] must be zero when n 1 = n 2 or τ 1 = τ 2 and E X [n 1 , τ 1 ]X[n 2 , τ 2 ] = E X [n 1 , τ 1 ]X[n 2 , τ 2 ] − E X [n 1 , τ 1 ] E X [n 2 , τ 2 ] = (U * J ΣU J )[(τ 1 − 1)N + n 1 , (τ 2 − 1)N + n 2 ]. Therefore, condition (b) is equivalent to stating that Σ = U J DU * J for some diagonal matrix D. In addition, (c) asserts that D[(τ −1)N +n, (τ −1)N +n] = h(λ n , ω τ ) for every n, τ .Thus taken together, (b) and (c) state that Σ = U J DU * J = U J h(Λ G , Λ T )U * J = h(L G , L T ), which is the second moment condition of a JWSS process.

Forecasting Time Series With VARMA Recursions on Graphs

“…In analogy with the definition of the JFT, stationarity can be extended to time-varying graph processes [16]. This new concept of stationarity requires now that the first and second order moments are jointly preserved along the graph and temporal dimension.…”

“…Continuing our prior works [16], [17], we exploit the (approximate) time-vertex stationarity of graph time series and extend classical VAR and vector autoregressive moving average (VARMA) recursions for modeling and predicting timevarying processes on graphs. Specifically, the contributions of this work are: 1) We propose VAR and VARMA models for forecasting time series on graphs.…”

“…Specifically, the contributions of this work are: 1) We propose VAR and VARMA models for forecasting time series on graphs. These recursions extend the noncausal models of [16] to causal ones and generalize the approach of [8] to (i) VAR recursions that do not impose a temporal interpretation of the edges, and (ii) VARMA recursions where the MA part allows to model time series with a more complete structure. 2) We propose a model fitting method that makes use of the data structure and stationarity.…”

“…The obtained results corroborate the relevance of graph VARMA models for forecasting stochastic time series supported on graphs and show that the proposed approaches outperform both classical univariate and multivariate ARMA techniques and state-of-the-art graph-based techniques. 2 We remark that although perfect stationarity either in the time or graph domain is difficult to be observed, several real data sets [13]- [16] have shown some degree of graph/time stationarity that can be exploited for prediction. 3 Data publicly available at https://donneespubliques.…”

Online Distributed Learning Over Graphs With Multitask Graph-Filter Models

“…In [34], vector autoregressive (VAR) and vector autoregressive moving average (VARMA) models are proposed for predicting time-varying processes on graphs. Joint time-vertex stationarity is introduced for time-varying graph signals in [35], [36], and a joint time-vertex harmonic analysis for graph signals is proposed in [37]. It is shown that joint stationarity facilitates estimation or recovery tasks when compared to purely time or graph methods.…”