Section: Research Program

Graph-based signal processing

Participants : Paulo Gonçalves, Rémi Gribonval, Marion Foare, Márton Karsai.

Evolving networks can be regarded as "out of equilibrium" systems.

Indeed, their dynamics are typically characterized by non standard and intricate statistical properties, such as non-stationarity, long range memory effects, intricate space and time correlations.

Analyzing, modeling, and even defining adapted concepts for dynamic graphs is at the heart of DANTE. This is a largely open question that has to be answered by keeping a balance between specificity (solutions triggered by specific data sets) and generality (universal approaches disconnected from social realities). We will tackle this challenge from a graph-based signal processing perspective involving signal analysts and computer scientists, together with experts of the data domain application. One can distinguish two different issues in this challenge, one related to the graph-based organization of the data and the other to the time dependency that naturally exits in the dynamic graph object. In both cases, a number of contributions can be found in the literature, albeit in different contexts. In our application domain, high-dimensional data "naturally reside" on the vertices of weighted graphs. The emerging field of signal processing on graphs merges algebraic and spectral graph theoretic concepts with computational harmonic analysis to process such signals on graphs  [74].

As for the first point, adapting well-founded signal processing techniques to data represented as graphs is an emerging, yet quickly developing field which has already received key contributions. Some of them are very general and delineate ambitious programs aimed at defining universal, generally unsupervised methods for exploring high-dimensional data sets and processing them. This is the case for instance of the “diffusion wavelets” and “diffusion maps” pushed forward at Yale and Duke  [57]. Others are more traditionally connected with standard signal processing concepts, in the spirit of elaborating new methodologies via some bridging between networks and time series, see for instance [69] and references therein. Other viewpoints can be found as well, including multi-resolution Markov models  [77], Bayesian networks or distributed processing over sensor networks  [68]. Such approaches can be particularly successful for handling static graphs and unveiling aspects of their organization in terms of dependencies between nodes, grouping, etc. Incorporating possible time dependencies within the whole picture calls however for the addition of an extra dimension to the problem "as it would be the case when switching from one image to a video sequence", a situation for which one can imagine to take advantage of the whole body of knowledge attached to non-stationary signal processing  [58].

The arrival of Rémi Gribonval in August 2019 brought a new dimension to the research program of this theme. Specialist of parsimonious representations of large data sets, R. Gribonval will develop at Dante a specific activity related to the sparsification of resources (computing and storage but also regarding the data volumes) in the context of machine and deep learning. This new orientation of Dante will be elaborated and fully integrated to the objectives of the future Inria project that will be proposed after Dante.