The lattices of closure systems, closure operators, and implicational systems on a finite set: a survey

Formal Concept Analysis (FCA) looks to decompose a matrix of objectsattributes into a set of sparse matrices capturing the underlying structure of a formal context. We propose a Rank Reduction (RR) method to prime approximate FCAs, namely RRFCA. While many existing FCA algorithms are complete, lectic ordering of the lattice may not minimize search/decomposition time. Initially, RRFCA decompositions are not unique or complete; however, a set of good closures with high support is learned quickly, and then, made complete. RRFCA has its novelty in that we propose a new multiplicative two-stage method. First, we describe the theoretical foundations underpinning our RR approach. Second, we provide a representative exemplar, showing how RRFCA can be implemented. Further experiments demonstrate that RRFCA methods are efficient, scalable and yield time-savings. We demonstrate the resulting methods lend themselves to parallelization.Key words: Formal Concept Analysis, Rank Reduction, Factorization. IntroductionFormal Concept Analysis (FCA) leverages the notion of a concept, an objectattribute building block of a binary relational dataset, and its ranking in a concept hierarchy to mine data-sets [25]. One short-coming is that concepts are mined according to lectic ordering and not concept importance or support in the formal context. Lectic ordering recommends itself on account of its thoroughness [10] [24,13]. The theoretical and empirical complexity of various approaches is compared by Kuznetsov in [16]. Computational complexity is the main measure for comparing algorithms: Kuznetsov and Obiedkov focus on the properties of the data ensemble, namely sparsity, the primary complexity inducing characteristic of the decomposition. Aside from sparsity, the main bottlenecks are memory and processing constraints. Ganter's algorithm computes concepts iteratively based on the previous concept, without incurring exponential memory requirements, by exploiting lectic ordering. CloseByOne produces many concepts in each iteration. Bordat's algorithm, described in [3], introduces a data structure to store previously found concepts, which results in considerable timesavings. This approach is made more efficient in [2] by removing the need for a structure of exponential size.A significant short-coming of batch approaches is that the entire lattice must be reconstructed if the database changes. Here, we look to the memory and computation challenge by using rank reduction method and disjointness to select good starting-intents for FCA.The justification goes as follows: all concepts are not equal in a binary relational dataset. FC support (the extent to which it overlaps with the formal context) and its expressiveness (FC disjointness given a set of FCs), may be different for FCs. NextClosure's lectic ordering does not consider these concerns. To address this, we prime NextClosure with multiple starting-intents by taking reduced rank approximations of the binary relation using Nonnegative Matrix Factorization (NMF) [18]. Lectic order...

Section: Contributionsmentioning

confidence: 99%

Formal Concept Analysis via Atomic Priming

Fréin

2013

“…In Formal Concept Analysis [15] and Galois Analysis [4], L is the powerset of a set P of attributes. Recent extensions to various languages have been performed [14,13].…”

Section: Extension-intension Latticesmentioning

confidence: 99%

“…In Formal Concept Analysis [15] and Galois lattice theory [4] L is a lattice, and the relation between L and the powerset P(O) is materialized as an extension-intension lattice. This lattice is the structure of the definable elements of P(O) [1], i.e.…”

Section: Introductionmentioning

confidence: 99%

“…The intuition here is a change in granularity: the new objects we deal with are the minimal abstractions of the original instances. Furthermore, as an extension-intension lattice represents a set of valid implications (see for instance [4]), our abstract extension-intension lattice also represents a set of valid abstract implications. Going back to our example, suppose ext(p) = {o 1 3 }, the abstract implication p → A q is valid.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Abstract Concept Lattices

Soldano

Ventos

2011

Abstract. We present a view of abstraction based on a structure preserving reduction of the Galois connection between a language L of terms and the powerset of a set of instances O. Such a relation is materialized as an extension-intension lattice, namely a concept lattice when L is the powerset of a set P of attributes. We define and characterize an abstraction A as some part of either the language or the powerset of O, defined in such a way that the extension-intension latticial structure is preserved. Such a structure is denoted for short as an abstract lattice. We discuss the extensional abstract lattices obtained by so reducing the powerset of O, together together with the corresponding abstract implications, and discuss alpha lattices as particular abstract lattices. Finally we give formal framework allowing to define a generalized abstract lattice whose language is made of terms mixing abstract and non abstract conjunctions of properties.

“…Formal Concept Analysis [1] as well as Galois analysis [2,3] relies on the relation between objects and attributes. In data mining, these ideas have been investigated under the name of itemsets mining and rely on the same relation [4].…”

Section: Introductionmentioning

confidence: 99%

Closed Patterns and Abstraction Beyond Lattices

Soldano¹

2014

Abstract. Recently pattern mining has investigated closure operators in families of subsets of an attribute set that are not lattices. In particular, various authors have investigated closure operators starting from a context, in the Formal Concept Analysis (FCA) sense, in which objects are described as usual according to their relation to attributes, and in which a closed element is a maximal element of the equivalence class of elements sharing the same support, i.e. occurring in the same objects. The purpose of this paper is twofold. First we thoroughly investigate this framework and relate it to FCA, defining in particular a structure called a preconfluence, weaker than a lattice, in which we can define a closure operator with respect to a set of objects. Second, we show that the requirements allowing us to define abstract concept lattices also allow us to define corresponding abstract Galois pre-confluences.