Maxime Bérar

Jason Piquenot, Aldo Moscatelli, Maxime Bérar et al.

This paper introduces a framework for formally establishing a connection between a portion of an algebraic language and a Graph Neural Network (GNN). The framework leverages Context-Free Grammars (CFG) to organize algebraic operations into generative rules that can be translated into a GNN layer model. As CFGs derived directly from a language tend to contain redundancies in their rules and variables, we present a grammar reduction scheme. By applying this strategy, we define a CFG that conforms to the third-order Weisfeiler-Lehman (3-WL) test using MATLANG. From this 3-WL CFG, we derive a GNN model, named G$^2$N$^2$, which is provably 3-WL compliant. Through various experiments, we demonstrate the superior efficiency of G$^2$N$^2$ compared to other 3-WL GNNs across numerous downstream tasks. Specifically, one experiment highlights the benefits of grammar reduction within our framework.

Mokhtar Z. Alaya, Maxime Bérar, Gilles Gasso et al.

We propose a novel approach for comparing distributions whose supports do not necessarily lie on the same metric space. Unlike Gromov-Wasserstein (GW) distance which compares pairwise distances of elements from each distribution, we consider a method allowing to embed the metric measure spaces in a common Euclidean space and compute an optimal transport (OT) on the embedded distributions. This leads to what we call a sub-embedding robust Wasserstein (SERW) distance. Under some conditions, SERW is a distance that considers an OT distance of the (low-distorted) embedded distributions using a common metric. In addition to this novel proposal that generalizes several recent OT works, our contributions stand on several theoretical analyses: (i) we characterize the embedding spaces to define SERW distance for distribution alignment; (ii) we prove that SERW mimics almost the same properties of GW distance, and we give a cost relation between GW and SERW. The paper also provides some numerical illustrations of how SERW behaves on matching problems.

Mokhtar Z. Alaya, Maxime Bérar, Gilles Gasso et al.

We introduce in this paper a novel strategy for efficiently approximating the Sinkhorn distance between two discrete measures. After identifying neglectable components of the dual solution of the regularized Sinkhorn problem, we propose to screen those components by directly setting them at that value before entering the Sinkhorn problem. This allows us to solve a smaller Sinkhorn problem while ensuring approximation with provable guarantees. More formally, the approach is based on a new formulation of dual of Sinkhorn divergence problem and on the KKT optimality conditions of this problem, which enable identification of dual components to be screened. This new analysis leads to the Screenkhorn algorithm. We illustrate the efficiency of Screenkhorn on complex tasks such as dimensionality reduction and domain adaptation involving regularized optimal transport.