## Code reproducing the experiments of the paper

There are two main MATLAB scripts included.

**experiment_JPSD_estimation.m**

The code tests the accuracy of JPSD estimation methods (mode 'accuracy') shown in Fig. 1 of the paper, as well as their computational complexity in machine time (mode 'complexity') shown in Fig. 2.**experiment_recovery.m**

The code tests recovery accuracy and generates the figures in Section VI B of the paper. It is possible to choose the dataset (choices: molene, pems, epidemic). In addition, the variable xaxis (miss, train) controls whether the x-axis of the figure will be the percentage of signal entries hidden from the algorithm, or the percentage of the total data used in training.

## 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 to 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.