Factorizing Boolean matrices using formal concepts and iterative usage of essential entries

Boolean matrices, i.e., matrices over a Boolean algebra, do not satisfy many of the fundamental properties of matrices over the real or complex field. Algebraic aspects of the Boolean pseudoinverse matrix concept are investigated. Maps operating on Boolean vector spaces by analogy with those on linear spaces are defined. Some properties of them are obtained. Most of these properties remain the same in the case of linear spaces and they are essential for solving linear systems. In particular orthogonal projections in linear spaces receive their algebraic analogue in Boolean vector spaces. The maps defined are useful for solving equations in the Boolean algebra, if no pseudoinverse matrix to a given matrix exists. We also share our experience in using computer algebra systems for teaching discrete mathematics and linear algebra. Some examples for computations with binary relations and Boolean matrices using computer algebra systems are given.

Factorization of matrices with grades via essential entries

Enumeration of Rosenberg-type hypercompositional structures defined by binary relations

Binary relations, matrices and inference developments

Formal concept analysis (FCA) is an important theory on the study of hierarchical structures caused by a binary relation between objects and attributes. And it is well known that a soft set always induces a binary relationship between objects and attributes for a formal context. Based on these facts, we study a new system called a soft context, which is induced using a soft set instead of a binary relation in the formal context, and investigate its theoretical foundations. First, we introduce the notions of soft context and soft concept for objects and study some basic properties. In particular, for the purpose of studying soft concepts, we introduce and study the notions of independent and dependent soft concepts. And we obtain the important things: (1) the set of all soft concepts can be decomposed into the set of independent soft concepts and the set of dependent soft concepts; (2) every dependent soft concept is generated by some independent soft concepts. Second, we obtain important properties about the relationship between formal context and soft contexts as the following: (1) each formal context always induces an associated soft context; (2) for a given formal context, there exists a bijection between the set of all formal concepts and the set of all soft concepts of the associated soft context induced by the given formal context. And we apply the above-mentioned properties to get the set of all formal concepts of a given formal context and explain that this method is more effective than the traditional way of computing all formal concepts in formal contexts using the example. Finally, we introduce the notion of soft concept lattice. Also, for a given formal context, we show that the formal concept lattice is order-isomorphic to the soft concept lattice of the associated soft context.

Recently, incomplete formal contexts have received more and more attention from the communities of formal concept analysis. Different from a complete context where the binary relations between all the objects and attribute are known, an incomplete formal context has at least a pair of object and attribute with a completely unknown binary relation. Partially known formal concepts use interval sets to indicate the incompleteness. Three-way formal concept analysis is capable of characterizing a target set by combining positive and negative attributes. However, how to describe target set, by pointing out what attributes it has with certainty and what attributes it has with possibility and what attributes it does not has with certainty and what attributes it does not has with possibility, is still an open problem. This paper combines the ideas of three-way formal concept analysis and partially known formal concepts and presents a framework of approximate three-way concept analysis. At first, approximate object-induced and attribute-induced three-way concept lattices are introduced, respectively. And then, the relationship between approximate three-way concept lattice and classical three-way concept lattice are investigated. Finally, examples are presented to demonstrate and verify the obtained results.

Extracting knowledge from huge data in a reasonable time is still a challenging problem. Most real data (structured or not) can be mapped to an equivalent binary context, with or without using a scaling method, as for extracting associations between words in a text, or in machine learning systems. In this paper, our objective is to find a minimal coverage of a relation [Formula: see text] with formal concepts. The problem is known to be NP-complete.1 In this paper, we exploit a particular difunctional relation embedded in any binary relation [Formula: see text], the fringe of [Formula: see text], to find an approximate conceptual coverage of [Formula: see text]. We use formal properties of fringes to find better algorithms calculating the minimal rectangular coverage of binary relation. Here, a formal context is considered as a binary relation. By exploiting some background on relational algebra in the present work, we merge some results of Belohlavek and Vichodyl,2 using formal concept analysis with previous results obtained by Kcherif et al.3 using relational algebra. We finally propose decomposition algorithms based on the relational formalization and fringe relations.

The storage of data is a key issue of information systems, which is an important foundation for data query and data mining. Relational database model has been proven to be a very useful data-storage technique. As information is stored as data in relational databases, the induction of concepts from data is a pivotal topic in the data mining field. Formal Concept Analysis (FCA) turns out to be a perfect instrument for a meaningful and conceptual exploration of the stored data. In FCA, conceptual scaling provides a complete framework for transforming any many-valued context (i.e., relation/table) into a context (called a derived context), in which each many valued attribute is given a scale. The attributes in a scale basically describe meaningful features of the values of the initial attribute. From the logical point of view, complement operation plays a very important role in relational databases and data query systems. In this paper, we provide the connections between the concepts of binary relations and those of complementary binary relations, and propose an approach toward normalizing (complementary) scales, i.e., each (complementary)scale can be represented by a set of statements. One advantage of normalizing scales is to avoid generating huge derived relations, and hence this approach reduces storage cost. By the normalization, the concept lattice of the complement of a derived relation is reduced to a combination of the concept lattice of the derived relation and a set of statements.

Formal Concept Analysis (FCA) provides an account of classification based on a binary relation between two sets. These two sets contain the objects and attributes (or properties) under consideration. In this paper I propose a generalization of formal concept analysis based on binary relations between hypergraphs, and more generally between pre-orders. A binary relation between any two sets already provides a bipartite graph, and this is a well-known perspective in FCA. However the use of graphs here is quite different as it corresponds to imposing extra structure on the sets of objects and of attributes. In the case of objects the resulting theory should provide a knowledge representation technique for structured collections of objects. The generalization is achieved by an application of work on mathematical morphology for hypergraphs.KeywordsBinary RelationFormal ConceptConcept LatticeRelation AlgebraIdentity RelationThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Binary relations are such a basic object that they appear in many places in mathematics and computer science. For instance, when dealing with graphs, program semantics, or termination guarantees, binary relations are always used at some point. In this survey, we focus on the relations themselves, and we consider algebraic and algorithmic questions. On the algebraic side, we want to understand and characterise the laws governing the behaviour of the following standard operations on relations: union, intersection, composition, converse, and reflexive-transitive closure. On the algorithmic side, we look for decision procedures for equality or inequality of relations. After having formally defined the calculus of relations, we recall the existing results about two well-studied fragments of particular importance: Kleene algebras and allegories. Unifying those fragments yields a decidable theory whose axiomatisability remains an open problem.

İlişkisel veri analizinde temel bir problem olan topluluk seziminde, ilişkisel veri için model seçimi hala açık bir problemdir. Bu bildiride, Bayesçi atama modelinin (BAM) genel atama çerçevesi içinde, karmaşık üyelik rastlantısal öbek modellerinin (KÜRÖB) model boyutunun tahmin edilmesi önerilmektedir. İlişkisel verilerin atama modelindeki Poisson sayıları olarak nasıl temsil edildiği açıklanmaktadır. Deney sonuçları sentetik ve gerçek veri kümeleri üzerinde gösterilmektedir. Genelleyici atama yaklaşımı, sadece model boyutunu değil, aynı zamanda en uygun ayrıştırmayı seçebilen genelleştirilmiş bir model seçimi çözümü vaat etmektedir

Habitat structure has been implicated as a source of bias for pitfall‐trap data but most evidence is observational or anecdotal. This study used an experimental approach to quantify biases due to habitat structure. In a randomized block design, I manipulated native grassland to create three types of habitat structure and measured pitfall‐trap catches of grassland ants. Small patches of modified habitat were surrounded by otherwise unmodified grassland with the assumption that population density remained unaffected by the modification and that the effects observed were due to changes in trappability. I assessed magnitude, direction, predictability, and consistency of bias for the following types of data: population abundance for single species, relative abundance among species, species composition of assemblages, and species richness. The magnitude of the bias in population abundance was large for most species. However, since the direction of the bias varied predictably with habitat structure, pitfall‐trap data can be used to judge differences in population abundance in some situations. The magnitude of the bias in relative abundance was less than for abundance. However, there was inconsistency in the direction and magnitude of bias among species. Thus, interpretation of relative abundance data in pitfall‐trap studies may be compromised. Species richness and species composition were biased by habitat structure but were affected significantly only when the groundcover was very dense, suggesting a threshold effect of habitat structure. To help to interpret survey data, pitfall‐trap studies should routinely measure attributes of habitat structure and incorporate an experimental component to characterize the bias.

This paper introduces a new multi-label classifier based on Boolean matrix decomposition. Boolean matrix decomposition is used to extract, from the full label matrix, latent labels representing useful Boolean combinations of the original labels. Base level models predict latent labels, which are subsequently transformed into the actual labels by Boolean matrix multiplication with the second matrix from the decomposition. The new method is tested on six publicly available datasets with varying numbers of labels. The experimental evaluation shows that the new method works particularly well on datasets with a large number of labels and strong dependencies among them.

We present a fast algorithm for computing the QR factorization of Cauchy-like matrices with real nodes. The algorithm is based on the existence of certain generalized recurrences among the columns of Q and RT, does not require squaring the matrix, and fully exploits tile displacement structure of Cauchy-like matrices. Moreover, we describe a class of Cauchy-like matrices admitting an explicit polar decomposition.

It is shown that the generalized Schur algorithm for triangular factorization of symmetric positive definite matrices has a natural extension to the factorization of symmetric indefinite matrices with nonsingular principal submatrices. The proof is constructive and provides for an explicit formulation of the J-orthogonal and triangular matrices involved in the procedure. The (group-theoretic) significance of degenerate transformation steps involving unbounded reflection coefficients is precisely identified. It is found how to assign them an interpretation as Schur parameters and how to get benefit from this knowledge for performing a suitable change of equivalence class during execution, instead of a breakdown of the algorithm. >