During the past few years, graph signal processing has been extending the field of signal processing on Euclidean spaces to irregular spaces represented by graphs. We have seen successes ranging from the Fourier transform, to wavelets, vertex-frequency (time-frequency) decomposition, sampling theory, uncertainty principle, or convolutive filtering. One missing ingredient though are the tools to study stochastic graph signals for which the randomness introduces its own difficulties. Classical signal processing has introduced a very simple yet very rich class of stochastic signals that is at the core of the study of stochastic signals: the stationary signals. These are the signals statistically invariant through a shift of the origin of time. In this talk, we study two extensions of stationarity to graph signals, one that stems from a new translation operator for graph signals, and another one with a more sensible interpretation on the graph. In the course, we show that attempts of alternate definitions of stationarity on graphs in the recent literature are actually equivalent to our first definition. Finally, we look at a real weather dataset and show empirical evidence of stationarity.

Bio: Benjamin Girault received his License (B.Sc.) and his Master (M.Sc.) in France from École Normale Supérieure de Cachan, France, in 2009 and 2012 respectively in the field of theoretical computer science. He then received his PhD in computer science from École Normale Supérieure de Lyon, France, in December 2015. His dissertation entitled "Signal Processing on Graphs - Contributions to an Emerging Field" focused on extending the classical definition of stationary temporal signals to stationary graph signal. Currently, he is a postdoctoral scholar with Antonio Ortega and Shri Narayanan at the University of Southern California continuing his work on graph signal processing with a focus on applying these tools to understanding human behavior.