A Convolutional Neural Network into graph space

Convolutional neural networks (CNNs), in a few decades, have outperformed the existing state of the art methods in classification context. However, in the way they were formalised, CNNs are bound to operate on euclidean spaces. Indeed, convolution is a signal operation that are defined on euclidean spaces. This has restricted deep learning main use to euclidean-defined data such as sound or image. And yet, numerous computer application fields (among which network analysis, computational social science, chemo-informatics or computer graphics) induce non-euclideanly defined data such as graphs, networks or manifolds. In this paper we propose a new convolution neural network architecture, defined directly into graph space. Convolution and pooling operators are defined in graph domain. We show its usability in a back-propagation context. Experimental results show that our model performance is at state of the art level on simple tasks. It shows robustness with respect to graph domain changes and improvement with respect to other euclidean and non-euclidean convolutional architectures.Comment: arXiv admin note: text overlap with arXiv:1611.08402 by other author

An Exact Graph Edit Distance Algorithm for Solving Pattern Recognition Problems

International audienceGraph edit distance is an error tolerant matching technique emerged as a powerful and flexible graph matching paradigm that can be used to address different tasks in pattern recognition, machine learning and data mining; it represents the minimum-cost sequence of basic edit operations to transform one graph into another by means of insertion, deletion and substitution of vertices and/or edges. A widely used method for exact graph edit distance computation is based on the A* algorithm. To overcome its high memory load while traversing the search tree for storing pending solutions to be explored, we propose a depth-first graph edit distance algorithm which requires less memory and searching time. An evaluation of all possible solutions is performed without explicitly enumerating them all. Candidates are discarded using an upper and lower bounds strategy. A solid experimental study is proposed; experiments on a publicly available database empirically demonstrated that our approach is better than the A* graph edit distance computation in terms of speed, accuracy and classification rate

Technical report: Graph Neural Networks go Grammatical

This paper proposes a framework to formally link a fragment of an algebraic language to a Graph Neural Network (GNN). It relies on Context Free Grammars (CFG) to organise algebraic operations into generative rules that can be translated into a GNN layer model. Since the rules and variables of a CFG directly derived from a language contain redundancies, a grammar reduction scheme is presented making tractable the translation into a GNN layer. Applying this strategy, a grammar compliant with the third-order Weisfeiler-Lehman (3-WL) test is defined from MATLANG. From this 3-WL CFG, we derive a provably 3-WL GNN model called G$^2$N$^2$. Moreover, this grammatical approach allows us to provide algebraic formulas to count the cycles of length up to six and chordal cycles at the edge level, which enlightens the counting power of 3-WL. Several experiments illustrate that G$^2$N$^2$ efficiently outperforms other 3-WL GNNs on many downstream tasks.Comment: 27 pages, 7 figure

From Kantorovitch Problem to Linear Sum Assignment Problem

The goal of this technical report is to detail the link between the Linear Sum Assignment Problem (LSAP) and the Kantorovitch Problem (KP) also called Optimal Transport. This relation is not new and is reported in \cite{peyre2020computational

Graph Mining and Graph Classification : application to cadastral map analysis

Les travaux présentés dans ce mémoire de thèse abordent sous différents angles très intéressants, un sujet vaste et ambitieux : l’interprétation de plans cadastraux couleurs.Dans ce contexte, notre approche se trouve à la confluence de différentes thématiques de recherche telles que le traitement du signal et des images, la reconnaissance de formes, l’intelligence artificielle et l’ingénierie des connaissances. En effet, si ces domaines scientifiques diffèrent dans leurs fondements, ils sont complémentaires et leurs apports respectifs sont indispensables pour la conception d’un système d’interprétation. Le centre du travail est le traitement automatique de documents cadastraux du 19e siècle. La problématique est traitée dans le cadre d'un projet réunissant des historiens, des géomaticiens et des informaticiens. D'une part nous avons considéré le problème sous un angle systémique, s'intéressant à toutes les étapes de la chaîne de traitements mais aussi avec un souci évident de développer des méthodologies applicables dans d'autres contextes. Les documents cadastraux ont été l'objet de nombreuses études mais nous avons su faire preuve d'une originalité certaine, mettant l'accent sur l'interprétation des documents et basant notre étude sur des modèles à base de graphes. Des propositions de traitements appropriés et de méthodologies ont été formulées. Le souci de comblé le gap sémantique entre l’image et l’interprétation a reçu dans le cas des plans cadastraux étudiés une réponse.This thesis tackles the problem of technical document interpretationapplied to ancient and colored cadastral maps. This subject is on the crossroadof different fields like signal or image processing, pattern recognition, artificial intelligence,man-machine interaction and knowledge engineering. Indeed, each of thesedifferent fields can contribute to build a reliable and efficient document interpretationdevice. This thesis points out the necessities and importance of dedicatedservices oriented to historical documents and a related project named ALPAGE.Subsequently, the main focus of this work: Content-Based Map Retrieval within anancient collection of color cadastral maps is introduced

A graph matching method based on leading Eigenvector and Sinkhorn-Knopp algorithm

The goal of the report is to present a graph matching method based on the leading Eigenvectorand Sinkhorn-Knopp algorithm. This method is not new and is reported in [6]

On the unification of the graph edit distance and graph matching problems

International audienceError-tolerant graph matching gathers an important family of problems. These problems aim at finding correspondences between two graphs while integrating an error model. In the Graph Edit Distance (GED) problem, the insertion/deletion of edges/nodes from one graph to another is explicitly expressed by the error model. At the opposite, the problem commonly referred to as "graph matching" does not explicitly express such operations. For decades, these two problems have split the research community in two separated parts. It resulted in the design of different solvers for the two problems. In this paper, we propose a unification of both problems thanks to a single model. We give the proof that the two problems are equivalent under a reformulation of the error models. This unification makes possible the use on both problems of existing solving methods from the two communities