A Model for Managing Collections of Patterns

Data mining algorithms are now able to efficiently deal with huge amount of data. Various kinds of patterns may be discovered and may have some great impact on the general development of knowledge. In many domains, end users may want to have their data mined by data mining tools in order to extract patterns that could impact their business. Nevertheless, those users are often overwhelmed by the large quantity of patterns extracted in such a situation. Moreover, some privacy issues, or some commercial one may lead the users not to be able to mine the data by themselves. Thus, the users may not have the possibility to perform many experiments integrating various constraints in order to focus on specific patterns they would like to extract. Post processing of patterns may be an answer to that drawback. Thus, in this paper we present a framework that could allow end users to manage collections of patterns. We propose to use an efficient data structure on which some algebraic operators may be used in order to retrieve or access patterns in pattern bases.

💡 Research Summary

The paper addresses the problem of managing large collections of patterns extracted by data‑mining tools, focusing on formal concepts as a representative pattern class. While existing approaches such as inductive databases and traditional Pattern Base Management Systems (PBMS) either provide only rudimentary post‑processing capabilities or rely on relational models that are ill‑suited for pattern queries, the authors propose a novel framework based on a labeled graph representation combined with algebraic operators.

A formal concept is defined as a maximal 1‑rectangle (X, Y) in a binary data matrix, where X is a set of attributes and Y a set of objects. Concepts are partially ordered by (X⊆X′, Y′⊆Y), forming a concept lattice. The authors map this lattice to a directed acyclic graph (the Hasse diagram) where each vertex corresponds to a concept and an edge exists only between a concept and its immediate successors (cover relation). Two special vertices, ⊤ (top) and ⊥ (bottom), are added to make the graph universal. Labels can be attached either to vertices (storing the full (X, Y) pair) or to edges (storing the differences X′\X and Y\Y′). This representation preserves the duality between attributes and objects and avoids any imposed ordering of attributes or objects.

The construction algorithm processes a list of concepts sorted by increasing |X|. For each new concept C=(X,Y), it initially connects C only to ⊤. A depth‑first traversal from ⊤ discovers all existing concepts that are covered by C; edges from those predecessors to C are added, and the obsolete edge from the predecessor to ⊤ is removed. The algorithm never explores sub‑graphs that cannot be covered by C, which yields an efficient O(|C|·|E|) construction time in practice.

Two families of queries are defined:

1. Selection (σₚ) – given a predicate p on concepts, σₚ(C) returns all (X,Y)∈C satisfying p. Typical predicates include minimum/maximum size of X, minimum/maximum support (|Y|), area constraints (|X|·|Y|), or membership of a specific attribute or object. Implemented as a simple scan of vertex labels, selection is linear in the number of concepts.

2. Projection (π_A) – given a subset A of attributes, the goal is to obtain the concept lattice of the projected database π_A(Db) without re‑mining the data. The authors introduce an A‑equivalence relation: two concepts are A‑equivalent iff X∩A are equal. Each equivalence class possesses a unique least element under the lattice order. The set LE_A of these least elements satisfies: Concepts(π_A(Db)) = { (X∩A, Y) | (X,Y) ∈ LE_A }. Thus, projection can be performed by grouping concepts according to X∩A, selecting the minimal element in each group, and trimming the attribute part to A. This operation is closed: the result is again a collection of concepts represented by the same graph structure, enabling chained queries.

The paper contrasts this approach with automaton‑based storage (prefix trees, minimal automata, commutative automata). Automata require an arbitrary ordering of attributes to encode sets as strings, which breaks the attribute/object duality and complicates queries. Moreover, commutative automata, while order‑independent, suffer from a large number of edges and store only attribute information, losing object data. The labeled graph avoids these drawbacks, offering a natural, order‑free representation that supports both attribute‑centric and object‑centric queries symmetrically.

Experimental discussion (though not detailed in the excerpt) indicates that the graph representation uses less memory than automaton‑based structures and that query execution times are competitive, especially for selection and projection operations that can be performed directly on the graph without touching the original data matrix.

In conclusion, the authors deliver a compact, duality‑preserving graph model for formal concept collections, together with a small algebra of selection and projection operators. This framework enables end‑users—who may lack direct access to raw data due to privacy or commercial constraints—to efficiently explore, filter, and transform large pattern bases. The approach is extensible to other pattern types (e.g., association rules, sequential patterns) and opens avenues for further research on richer algebraic operators and optimized graph construction algorithms.