Event detection and structure inference for graph vectors.

Summary This thesis addresses different problems around the analysis and modeling of signals on graphs, in other words vector data observed on graphs. We are particularly interested in two specific tasks. The first one is the problem of event detection, i.e. the detection of anomalies or breaks, in a set of vectors on graphs. The second task consists in the inference of the graph structure underlying the vectors contained in a data set. At first, our work is application oriented. We propose a method to detect antenna failures in a telecommunication network. The proposed methodology is designed to be efficient for communication networks in a broad sense and implicitly takes into account the underlying structure of the data. In a second step, a new graph inference method in the framework of Graph Signal Processing is studied. In this problem, notions of local and global regularity, with respect to the underlying graph, are imposed on vectors. Finally, we propose to combine the graph learning task with the break detection problem. This time, a probabilistic framework is considered to model the vectors, assumed to be distributed according to a certain Markov random field. In our modeling, the graph underlying the data can change over time and a breakpoint is detected whenever it changes significantly.