## Section: New Results

### Graph Signal Processing and Machine Learning

Participants : Paulo Gonçalves, Rémi Gribonval, Marion Foare, Thomas Begin, Esteban Bautista Ruiz, Gaetan Frusque, Amélie Barbe, Mikhail Tsitsvero, Marija Stojanova, Márton Karsai, Sébastien Lerique, Jacobo Levy Abitbol.

#### ${L}^{\gamma}$ -PageRank for Semi-Supervised Learning

Participants : Paulo Gonçalves, Esteban Bautista Ruiz.

PageRank for Semi-Supervised Learning has shown to leverage data structures and limited tagged examples to yield meaningful classification. Despite successes, classification performance can still be improved, particularly in cases of fuzzy graphs or unbalanced labeled data. To address such limitations, a novel approach based on powers of the Laplacian matrix ${L}^{\gamma}$ ($\gamma >0$), referred to as ${L}^{\gamma}$-PageRank, is proposed. Its theoretical study shows that it operates on signed graphs, where nodes belonging to one same class are more likely to share positive edges while nodes from different classes are more likely to be connected with negative edges. It is shown that by selecting an optimal $\gamma $, classification performance can be significantly enhanced. A procedure for the automated estimation of the optimal $\gamma $, from a unique observation of data, is devised and assessed. Experiments on several datasets demonstrate the effectiveness of both ${L}^{\gamma}$-PageRank classification and the optimal $\gamma $ estimation. [11]

#### Designing Convex Combination of Graph Filters

Participant : Paulo Gonçalves.

In this work, we studied the problem of parametric modeling of network-structured signals with graph filters. Unlike the popular polynomial graph filters, which are based on a single graph shift operator, we considered convex combinations of graph shift operators particularly adapted to directed graphs. As the resulting modeling problem is not convex, we reformulated it as a convex optimization problem which can be solved efficiently. Experiments on real-world data structured by undirected and directed graphs were conducted. The results showed the effectiveness of this method compared to other methods reported in the literature. [18]

#### Optimal transport under regularity constraints for domain adaptation between graphs with attributes

Participants : Paulo Gonçalves, Amélie Barbe.

In this work, we addresses the problem of domain adaptation between two graphs by optimal transport. We aimed at benefiting from the knowledge of a labeled source graph to improve the classification of nodes in an unlabeled target graph.
We focused on the setting where a set of features is associated to each node of the graphs.
We proposed an original method that optimizes a transportation plan from the source to the target that *(i)* preserves the structures transported between the graphs and *(ii)* prevents the mapping from transporting two source nodes with different labels to the same destination.
[30]

#### Sparse tensor dimensionality reduction with application to the clustering of functional connectivity in the brain

Participants : Paulo Gonçalves, Gaetan Frusque.

Functional connectivity (FC) is a graph-like data structure commonly used by neuroscientists to study the dynamic behaviour of the brain activity. However, these analyses rapidly become complex and time-consuming, as the number of connectivity components to be studied is quadratic with the number of electrodes. In our work, we addressed the problem of clustering FC into relevant ensembles of simultaneously activated components that reveal characteristic patterns of the epileptic seizures of a given patient. While $k-$means is certainly the most popular method for data clustering, it is known to perform badly on large dimensional data sets, and to be highly sensitive to noise. To overcome the co-called curse of dimensionality, we proposed a new tensor decomposition to reduce the size of the data set formed by FC time series recorded for several seizures, before applying $k$-means. Our contribution is twofold: First, we derived a method that we compared to the state of the art, emphasizing one variant that imposes sparsity constraints. Second, we conducted a real case study, applying the proposed sparse tensor decomposition to epileptic data in order to infer the functional connectivity graph dynamics corresponding to the different stages of an epileptic seizure. [31], [47]

#### Graph signal processing to model WLANs performances

Participants : Paulo Gonçalves, Thomas Begin, Marija Stojanova.

As WLANs have become part of our everyday life, there is an increasing need for more transmission capacity and wireless coverage. In response to this growing need, network administrators tend to intensify the deployment of Access Points (APs). However, if not correctly done, this AP densification may lead to badly planned and uncoordinated networks with sub-optimal use of the available resources. In this work, we propose a data-driven approach using graph signal processing and a set of input/output signals to capture the behavior of a WLAN and derive a predictive performance model. Given the simplicity and the novelty of the proposed model, we believe that its relative error of around 10-20% in modeling and 25% in prediction may represent a promising start for new approaches in the modeling of WLANs. [33]

#### Joint embedding of structure and features via graph convolutional networks

Participants : Márton Karsai, Sébastien Lerique.

We propose *AN2VEC*, a node embedding method which ultimately aims at disentangling the information shared by the structure of a network and the features of its nodes. Building on the recent developments of Graph Convolutional Networks (GCN), we develop a multitask GCN Variational Autoencoder where different dimensions of the generated embeddings can be dedicated to encoding feature information, network structure, and shared feature-network information. We explore the interaction between these disentangled characters by comparing the embedding reconstruction performance to a baseline case where no shared information is extracted. We use synthetic datasets with different levels of interdependency between feature and network characters and show (i) that shallow embeddings relying on shared information perform better than the corresponding reference with unshared information, (ii) that this performance gap increases with the correlation between network and feature structure, and (iii) that our embedding is able to capture joint information of structure and features. Our method can be relevant for the analysis and prediction of any featured network structure ranging from online social systems to network medicine.
[51]