Section: New Results

Graph & Signal Processing

Participants : Paulo Gonçalves, Éric Fleury, Sarra Ben Alaya, Esteban Bautista Ruiz, Gaëtan Frusque, Sarah de Nigris, Mikhail Tsitsvero.

Fractional Semi-Supervised Machine Learning

Graph-based semi-supervised learning for classification endorses a nice interpretation in terms of diffusive random walks, where the regularisation factor in the original optimisation formulation plays the role of a restarting probability. Recently, a new type of biased random walks for characterising certain dynamics on networks have been defined and rely on the $\gamma$-th power of the standard Laplacian matrix ${𝐋}^{\gamma }$, with $\gamma >0$. In particular, these processes embed long range transitions, the Lévy flights, that are capable of one-step jumps between far-distant states (nodes) of the graph. In a series of two articles [28] and [29], we envisioned to build upon these volatile random walks to propose two new versions of graph based semi-supervised learning algorithms: one called fractional SSL corresponds to the case where $0<\gamma <1$ whose classification outcome could benefit from the dynamics induced by the fractional transition matrix, and the other less straightforwardly connected to random walks, derives from $\gamma >1$.

Design of graph filters and filterbanks

Basic operations in graph signal processing consist in processing signals indexed on graphs either by filtering them or by changing their domain of representation, in order to better extract or analyze the important information they contain. The aim of our chapter [58] is to review general concepts underlying such filters and representations of graph signals. We first recall the different Graph Fourier Transforms that have been developed in the literature, and show how to introduce a notion of frequency analysis for graph signals by looking at their variations. Then, we move to the introduction of graph filters, that are defined like the classical equivalent for 1D signals or 2D images, as linear systems which operate on each frequency of a signal. Some examples of filters and of their implementations are given. Finally, as alternate representations of graph signals, we focus on multiscale transforms that are defined from filters. Continuous multiscale transforms such as spectral wavelets on graphs are reviewed, as well as the versatile approaches of filterbanks on graphs. Several variants of graph filterbanks are discussed, for structured as well as arbitrary graphs, with a focus on the central point of the choice of the decimation or aggregation operators.

GraSP: A Matlab Toolbox for Graph Signal Processing

In [30], we publicised the recent developments and new functionalities of our Graph Signal Processing Toolbox (GraSP).