Maximal Ordinal Two-Factorizations

Given a formal context, an ordinal factor is a subset of its incidence relation that forms a chain in the concept lattice, i.e., a part of the dataset that corresponds to a linear order. To visualize the data in a formal context, Ganter and Glodeanu proposed a biplot based on two ordinal factors. For the biplot to be useful, it is important that these factors comprise as much data points as possible, i.e., that they cover a large part of the incidence relation. In this work, we investigate such ordinal two-factorizations. First, we investigate for formal contexts that omit ordinal two-factorizations the disjointness of the two factors. Then, we show that deciding on the existence of two-factorizations of a given size is an NP-complete problem which makes computing maximal factorizations computationally expensive. Finally, we provide the algorithm Ord2Factor that allows us to compute large ordinal two-factorizations.Comment: 15 pages, 6 figures, 2 algorithms, 28th International Conference on Conceptual Structure

RV-Xplorer: A Way to Navigate Lattice-Based Views over RDF Graphs

International audienceMore and more data are being published in the form of machine readable RDF graphs over Linked Open Data (LOD) Cloud accessible through SPARQL queries. This study provides interactive navigation of RDF graphs obtained by SPARQL queries using Formal Concept Analysis. With the help of this {\tt View By} clause a concept lattice is created as an answer to the SPARQL query which can then be visualized and navigated using RV-Xplorer (Rdf View eXplorer). Accordingly, this paper discusses the support provided to the expert for answering certain questions through the navigation strategies provided by RV-Xplorer. Moreover, the paper also provides a comparison existing state of the art approaches

Analysis of social communities with iceberg and stability-based concept lattices

International audienceIn this paper, we presents a research work based on formal concept analysis and interest measures associated with formal concepts. This work focuses on the ability of concept lattices to discover and represent special groups of individuals, called social communities. Concept lattices are very useful for the task of knowledge discovery in databases, but they are hard to analyze when their size become too large. We rely on concept stability and support measures to reduce the size of large concept lattices. We propose an example from real medical use cases and we discuss the meaning and the interest of concept stability for extracting and explaining social communities within a healthcare network

The Singular Value Decomposition over Completed Idempotent Semifields

In this paper, we provide a basic technique for Lattice Computing: an analogue of the Singular Value Decomposition for rectangular matrices over complete idempotent semifields (i-SVD). These algebras are already complete lattices and many of their instances—the complete schedule algebra or completed max-plus semifield, the tropical algebra, and the max-times algebra—are useful in a range of applications, e.g., morphological processing. We further the task of eliciting the relation between i-SVD and the extension of Formal Concept Analysis to complete idempotent semifields (K-FCA) started in a prior work. We find out that for a matrix with entries considered in a complete idempotent semifield, the Galois connection at the heart of K-FCA provides two basis of left- and right-singular vectors to choose from, for reconstructing the matrix. These are join-dense or meet-dense sets of object or attribute concepts of the concept lattice created by the connection, and they are almost surely not pairwise orthogonal. We conclude with an attempt analogue of the fundamental theorem of linear algebra that gathers all results and discuss it in the wider setting of matrix factorization.This research was funded by the Spanish Government-MinECo project TEC2017-84395-P and the Dept. of Research and Innovation of Madrid Regional Authority project EMPATIA-CM (Y2018/TCS-5046)

Automatic Construction of Implicative Theories for Mathematical Domains

Implication is a logical connective corresponding to the rule of causality "if ... then ...". Implications allow one to organize knowledge of some field of application in an intuitive and convenient manner. This thesis explores possibilities of automatic construction of all valid implications (implicative theory) in a given field. As the main method for constructing implicative theories a robust active learning technique called Attribute Exploration was used. Attribute Exploration extracts knowledge from existing data and offers a possibility of refining this knowledge via providing counter-examples. In frames of the project implicative theories were constructed automatically for two mathematical domains: algebraic identities and parametrically expressible functions. This goal was achieved thanks both pragmatical approach of Attribute Exploration and discoveries in respective fields of application. The two diverse application fields favourably illustrate different possible usage patterns of Attribute Exploration for automatic construction of implicative theories

Towards Ordinal Data Science

Order is one of the main instruments to measure the relationship between objects in (empirical) data. However, compared to methods that use numerical properties of objects, the amount of ordinal methods developed is rather small. One reason for this is the limited availability of computational resources in the last century that would have been required for ordinal computations. Another reason - particularly important for this line of research - is that order-based methods are often seen as too mathematically rigorous for applying them to real-world data. In this paper, we will therefore discuss different means for measuring and ‘calculating’ with ordinal structures - a specific class of directed graphs - and show how to infer knowledge from them. Our aim is to establish Ordinal Data Science as a fundamentally new research agenda. Besides cross-fertilization with other cornerstone machine learning and knowledge representation methods, a broad range of disciplines will benefit from this endeavor, including, psychology, sociology, economics, web science, knowledge engineering, scientometrics

Interrogation d'un réseau sémantique de documents : l'intertextualité dans l'accès à l'information juridique

A collection of documents is generally represented as a set of documents but this simple representation does not take into account cross references between documents, which often defines their context of interpretation. This standard document model is less adapted for specific professional uses in specialized domains in which documents are related by many various references and the access tools need to consider this complexity. We propose two models based on formal and relational concept analysis and on semantic web techniques. Applied on documentary objects, these two models represent and query in a unified way documents content descriptors and documents relations.Une collection documentaire est généralement représentée comme un ensemble de documents mais cette modélisation ne permet pas de rendre compte des relations intertextuelles et du contexte d'interprétation d'un document. Le modèle documentaire classique trouve ses limites dans les domaines spécialisés où les besoins d'accès à l'information correspondent à des usages spécifiques et où les documents sont liés par de nombreux types de relations. Ce travail de thèse propose deux modèles permettant de prendre en compte cette complexité des collections documentaire dans les outils d'accès à l'information. Le premier modèle est basée sur l'analyse formelle et relationnelle de concepts, le deuxième est basée sur les technologies du web sémantique. Appliquées sur des objets documentaires ces modèles permettent de représenter et d'interroger de manière unifiée les descripteurs de contenu des documents et les relations intertextuelles qu'ils entretiennent