Revisiting Numerical Pattern Mining with Formal Concept Analysis

In this paper, we investigate the problem of mining numerical data in the framework of Formal Concept Analysis. The usual way is to use a scaling procedure --transforming numerical attributes into binary ones-- leading either to a loss of information or of efficiency, in particular w.r.t. the volume of extracted patterns. By contrast, we propose to directly work on numerical data in a more precise and efficient way, and we prove it. For that, the notions of closed patterns, generators and equivalent classes are revisited in the numerical context. Moreover, two original algorithms are proposed and used in an evaluation involving real-world data, showing the predominance of the present approach

Recursive Solution of Initial Value Problems with Temporal Discretization

We construct a continuous domain for temporal discretization of differential equations. By using this domain, and the domain of Lipschitz maps, we formulate a generalization of the Euler operator, which exhibits second-order convergence. We prove computability of the operator within the framework of effectively given domains. The operator only requires the vector field of the differential equation to be Lipschitz continuous, in contrast to the related operators in the literature which require the vector field to be at least continuously differentiable. Within the same framework, we also analyze temporal discretization and computability of another variant of the Euler operator formulated according to Runge-Kutta theory. We prove that, compared with this variant, the second-order operator that we formulate directly, not only imposes weaker assumptions on the vector field, but also exhibits superior convergence rate. We implement the first-order, second-order, and Runge-Kutta Euler operators using arbitrary-precision interval arithmetic, and report on some experiments. The experiments confirm our theoretical results. In particular, we observe the superior convergence rate of our second-order operator compared with the Runge-Kutta Euler and the common (first-order) Euler operators.Comment: 50 pages, 6 figure

Formal Concept Analysis Applications in Bioinformatics

Bioinformatics is an important field that seeks to solve biological problems with the help of computation. One specific field in bioinformatics is that of genomics, the study of genes and their functions. Genomics can provide valuable analysis as to the interaction between how genes interact with their environment. One such way to measure the interaction is through gene expression data, which determines whether (and how much) a certain gene activates in a situation. Analyzing this data can be critical for predicting diseases or other biological reactions. One method used for analysis is Formal Concept Analysis (FCA), a computing technique based in partial orders that allows the user to examine the structural properties of binary data based on which subsets of the data set depend on each other. This thesis surveys, in breadth and depth, the current literature related to the use of FCA for bioinformatics, with particular focus on gene expression data. This includes descriptions of current data management techniques specific to FCA, such as lattice reduction, discretization, and variations of FCA to account for different data types. Advantages and shortcomings of using FCA for genomic investigations, as well as the feasibility of using FCA for this application are addressed. Finally, several areas for future doctoral research are proposed. Adviser: Jitender S. Deogu

Anytime Subgroup Discovery in Numerical Domains with Guarantees

International audienceSubgroup discovery is the task of discovering patterns that accurately discriminate a class label from the others. Existing approaches can uncover such patterns either through an exhaustive or an approximate exploration of the pattern search space. However, an exhaustive exploration is generally unfeasible whereas approximate approaches do not provide guarantees bounding the error of the best pattern quality nor the exploration progression ("How far are we of an exhaustive search"). We design here an algorithm for mining numerical data with three key properties w.r.t. the state of the art: (i) It yields progressively interval patterns whose quality improves over time; (ii) It can be interrupted anytime and always gives a guarantee bounding the error on the top pattern quality and (iii) It always bounds a distance to the exhaustive exploration. After reporting experimentations showing the effectiveness of our method, we discuss its generalization to other kinds of patterns

A heuristic for boundedness of ranks of elliptic curves

We present a heuristic that suggests that ranks of elliptic curves over the rationals are bounded. In fact, it suggests that there are only finitely many elliptic curves of rank greater than 21. Our heuristic is based on modeling the ranks and Shafarevich-Tate groups of elliptic curves simultaneously, and relies on a theorem counting alternating integer matrices of specified rank. We also discuss analogues for elliptic curves over other global fields.Comment: 41 pages, typos fixed in torsion table in section

Proceedings of the 5th International Workshop "What can FCA do for Artificial Intelligence?", FCA4AI 2016(co-located with ECAI 2016, The Hague, Netherlands, August 30th 2016)

International audienceThese are the proceedings of the fifth edition of the FCA4AI workshop (http://www.fca4ai.hse.ru/). Formal Concept Analysis (FCA) is a mathematically well-founded theory aimed at data analysis and classification that can be used for many purposes, especially for Artificial Intelligence (AI) needs. The objective of the FCA4AI workshop is to investigate two main main issues: how can FCA support various AI activities (knowledge discovery, knowledge representation and reasoning, learning, data mining, NLP, information retrieval), and how can FCA be extended in order to help AI researchers to solve new and complex problems in their domain. Accordingly, topics of interest are related to the following: (i) Extensions of FCA for AI: pattern structures, projections, abstractions. (ii) Knowledge discovery based on FCA: classification, data mining, pattern mining, functional dependencies, biclustering, stability, visualization. (iii) Knowledge processing based on concept lattices: modeling, representation, reasoning. (iv) Application domains: natural language processing, information retrieval, recommendation, mining of web of data and of social networks, etc

Cluster simulation of relativistic fermions in two space-time dimensions

For Majorana-Wilson lattice fermions in two dimensions we derive a dimer representation. This is equivalent to Gattringer's loop representation, but is made exact here on the torus. A subsequent dual mapping leads to yet another representation in which a highly efficient Swendsen-Wang type cluster algorithm is constructed. It includes the possibility of fluctuating boundary conditions. It also allows for improved estimators and makes interesting new observables accessible to Monte Carlo. The algorithm is compatible with the Gross-Neveu as well as an additional Z(2) gauge interaction. In this article numerical demonstrations are reported for critical free fermions.Comment: 24 pages, 3 figures: tiny changes, mainly typo