Mireille El Gheche

Isabela Cunha Maia Nobre, Mireille El Gheche, Pascal Frossard

Graph learning is often a necessary step in processing or representing structured data, when the underlying graph is not given explicitly. Graph learning is generally performed centrally with a full knowledge of the graph signals, namely the data that lives on the graph nodes. However, there are settings where data cannot be collected easily or only with a non-negligible communication cost. In such cases, distributed processing appears as a natural solution, where the data stays mostly local and all processing is performed among neighbours nodes on the communication graph. We propose here a novel distributed graph learning algorithm, which permits to infer a graph from signal observations on the nodes under the assumption that the data is smooth on the target graph. We solve a distributed optimization problem with local projection constraints to infer a valid graph while limiting the communication costs. Our results show that the distributed approach has a lower communication cost than a centralised algorithm without compromising the accuracy in the inferred graph. It also scales better in communication costs with the increase of the network size, especially for sparse networks.

Hermina Petric Maretic, Mireille El Gheche, Giovanni Chierchia et al.

Graph comparison deals with identifying similarities and dissimilarities between graphs. A major obstacle is the unknown alignment of graphs, as well as the lack of accurate and inexpensive comparison metrics. In this work we introduce the filter graph distance. It is an optimal transport based distance which drives graph comparison through the probability distribution of filtered graph signals. This creates a highly flexible distance, capable of prioritising different spectral information in observed graphs, offering a wide range of choices for a comparison metric. We tackle the problem of graph alignment by computing graph permutations that minimise our new filter distances, which implicitly solves the graph comparison problem. We then propose a new approximate cost function that circumvents many computational difficulties inherent to graph comparison and permits the exploitation of fast algorithms such as mirror gradient descent, without grossly sacrificing the performance. We finally propose a novel algorithm derived from a stochastic version of mirror gradient descent, which accommodates the non-convexity of the alignment problem, offering a good trade-off between performance accuracy and speed. The experiments on graph alignment and classification show that the flexibility gained through filter graph distances can have a significant impact on performance, while the difference in speed offered by the approximation cost makes the framework applicable in practical settings.

Mireille El Gheche, Pascal Frossard

We are interested in multilayer graph clustering, which aims at dividing the graph nodes into categories or communities. To do so, we propose to learn a clustering-friendly embedding of the graph nodes by solving an optimization problem that involves a fidelity term to the layers of a given multilayer graph, and a regularization on the (single-layer) graph induced by the embedding. The fidelity term uses the contrastive loss to properly aggregate the observed layers into a representative embedding. The regularization pushes for a sparse and community-aware graph, and it is based on a measure of graph sparsification called "effective resistance", coupled with a penalization of the first few eigenvalues of the representative graph Laplacian matrix to favor the formation of communities. The proposed optimization problem is nonconvex but fully differentiable, and thus can be solved via the descent gradient method. Experiments show that our method leads to a significant improvement w.r.t. state-of-the-art multilayer graph clustering algorithms.

Mireille El Gheche, Pascal Frossard

Multilayer graphs are appealing mathematical tools for modeling multiple types of relationship in the data. In this paper, we aim at analyzing multilayer graphs by properly combining the information provided by individual layers, while preserving the specific structure that allows us to eventually identify communities or clusters that are crucial in the analysis of graph data. To do so, we learn a clustered representative graph by solving an optimization problem that involves a data fidelity term to the observed layers, and a regularization pushing for a sparse and community-aware graph. We use the contrastive loss as a data fidelity term, in order to properly aggregate the observed layers into a representative graph. The regularization is based on a measure of graph sparsification called "effective resistance", coupled with a penalization of the first few eigenvalues of the representative graph Laplacian matrix to favor the formation of communities. The proposed optimization problem is nonconvex but fully differentiable, and thus can be solved via the projected gradient method. Experiments show that our method leads to a significant improvement w.r.t. state-of-the-art multilayer graph learning algorithms for solving clustering problems.

Hermina Petric Maretic, Mireille El Gheche, Matthias Minder et al.

We propose a novel method for comparing non-aligned graphs of different sizes, based on the Wasserstein distance between graph signal distributions induced by the respective graph Laplacian matrices. Specifically, we cast a new formulation for the one-to-many graph alignment problem, which aims at matching a node in the smaller graph with one or more nodes in the larger graph. By integrating optimal transport in our graph comparison framework, we generate both a structurally-meaningful graph distance, and a signal transportation plan that models the structure of graph data. The resulting alignment problem is solved with stochastic gradient descent, where we use a novel Dykstra operator to ensure that the solution is a one-to-many (soft) assignment matrix. We demonstrate the performance of our novel framework on graph alignment and graph classification, and we show that our method leads to significant improvements with respect to the state-of-the-art algorithms for each of these tasks.

Hermina Petric Maretic, Mireille EL Gheche, Giovanni Chierchia et al.

We present a novel framework based on optimal transport for the challenging problem of comparing graphs. Specifically, we exploit the probabilistic distribution of smooth graph signals defined with respect to the graph topology. This allows us to derive an explicit expression of the Wasserstein distance between graph signal distributions in terms of the graph Laplacian matrices. This leads to a structurally meaningful measure for comparing graphs, which is able to take into account the global structure of graphs, while most other measures merely observe local changes independently. Our measure is then used for formulating a new graph alignment problem, whose objective is to estimate the permutation that minimizes the distance between two graphs. We further propose an efficient stochastic algorithm based on Bayesian exploration to accommodate for the non-convexity of the graph alignment problem. We finally demonstrate the performance of our novel framework on different tasks like graph alignment, graph classification and graph signal prediction, and we show that our method leads to significant improvement with respect to the-state-of-art algorithms.

Mireille El Gheche, Giovanni Chierchia, Pascal Frossard

In this paper, we propose a scalable algorithm for spectral embedding. The latter is a standard tool for graph clustering. However, its computational bottleneck is the eigendecomposition of the graph Laplacian matrix, which prevents its application to large-scale graphs. Our contribution consists of reformulating spectral embedding so that it can be solved via stochastic optimization. The idea is to replace the orthogonality constraint with an orthogonalization matrix injected directly into the criterion. As the gradient can be computed through a Cholesky factorization, our reformulation allows us to develop an efficient algorithm based on mini-batch gradient descent. Experimental results, both on synthetic and real data, confirm the efficiency of the proposed method in term of execution speed with respect to similar existing techniques.

Mireille El Gheche, Giovanni Chierchia, Pascal Frossard

Network data appears in very diverse applications, like biological, social, or sensor networks. Clustering of network nodes into categories or communities has thus become a very common task in machine learning and data mining. Network data comes with some information about the network edges. In some cases, this network information can even be given with multiple views or multiple layers, each one representing a different type of relationship between the network nodes. Increasingly often, network nodes also carry a feature vector. We propose in this paper to extend the node clustering problem, that commonly considers only the network information, to a problem where both the network information and the node features are considered together for learning a clustering-friendly representation of the feature space. Specifically, we design a generic two-step algorithm for multilayer network data clustering. The first step aggregates the different layers of network information into a graph representation given by the geometric mean of the network Laplacian matrices. The second step uses a neural net to learn a feature embedding that is consistent with the structure given by the network layers. We propose a novel algorithm for efficiently training the neural net via stochastic gradient descent, which encourages the neural net outputs to span the leading eigenvectors of the aggregated Laplacian matrix, in order to capture the pairwise interactions on the network, and provide a clustering-friendly representation of the feature space. We demonstrate with an extensive set of experiments on synthetic and real datasets that our method leads to a significant improvement w.r.t. state-of-the-art multilayer graph clustering algorithms, as it judiciously combines nodes features and network information in the node embedding algorithms.