Learning Parametric Time-Vertex Graph Processes from Incomplete Realizations

We consider the problem of estimating time-varying graph signals with missing observations, which is of interest in many applications involving data acquisition on irregular topologies. We model time-varying graph signals as jointly stationary time-vertex ARMA graph processes. We formulate the learning of ARMA process parameters as an optimization problem where the joint power spectral density of the model is fit to a rough empirical estimate of the process covariance matrix. We propose a convex relaxation of this problem, which results in an algorithm more flexible than existing methods regarding the pattern of available and missing observations of the process. Experimental results on meteorological signals show that the proposed method compares favorably to reference state-of-the-art algorithms.

1. Introduction

The inference of data on networks is currently a problem of interest in many applications. Data collections on irregular topologies such as sensor networks or social networks can typically be modeled as graph signals. In many scenarios, data measurements may vary over time; for instance, the measurements on a sensor network, or the tendencies of users on a social network may have time-dependent characteristics. These time-varying measurements can be modeled as time-varying graph signals. Meanwhile, in many data acquisition scenarios over networks, measurements are only partially observed in the vertex domain and the time domain, due to e.g., sensor failures, partial availability of user information, such that measurements may be missing at arbitrary graph nodes at arbitrary time instants. The inference of the unobserved measurements from the observed ones is a problem relevant to many applications. In this work, we consider the problem of estimating time-varying graph signals from partial observations without any assumptions on the observation pattern, i.e., allowing the missing observations to occur at any graph nodes and any time instants.