Computing the Concept Lattice using Dendritical Neural Networks

Formal Concept Analysis (FCA) is a new and rich emerging discipline, and it provides e cient techniques and methods for e cient data analysis under the idea of attributes . The main tool used in this area is the Concept Lattice also named Galois Lattice or Maximal Rectangle Lattice. A naive way to generate the Concept Lattice is by enumeration of each cluster of attributes. Unfortunately the numbers of clusters under the inclusion attribute relation has an exponential upper bound. In this work, we present a novel algorithm, PIRA (PIRA is a Recursive Acronym), for computing Concept Lattices in an elegant way. This task is achieved through the relation between maximal height and width rectangles, and maximal anti-chains. Then, using a dendritical neural network is possible to identify the maximal anti-chains in the lattice structure by means of maximal height or width rectangles.

[1]  Brian A. Davey,et al.  An Introduction to Lattices and Order , 1989 .

[2]  Manuel Graña,et al.  A Novel Lattice Associative Memory Based on Dendritic Computing , 2012, HAIS.

[3]  Isabelle Bloch,et al.  Mathematical Morphology Operators over Concept Lattices , 2013, ICFCA.

[4]  Lhouari Nourine,et al.  A fast incremental algorithm for building lattices , 2002, J. Exp. Theor. Artif. Intell..

[5]  Rokia Missaoui,et al.  A partition-based approach towards constructing Galois (concept) lattices , 2002, Discret. Math..

[6]  Rudolf Wille,et al.  Restructuring Lattice Theory: An Approach Based on Hierarchies of Concepts , 2009, ICFCA.

[7]  Etienne E. Kerre,et al.  A Lattice-Based Approach to Mathematical Morphology for Greyscale and Colour Images , 2007, Computational Intelligence Based on Lattice Theory.

[8]  Gerhard X. Ritter,et al.  Computational Intelligence Based on Lattice Theory , 2007, Studies in Computational Intelligence.

[9]  Gerhard X. Ritter,et al.  Single Layer Morphological Perceptron Solution to the N-Bit Parity Problem , 2004, CIARP.

[10]  Lhouari Nourine,et al.  A Fast Algorithm for Building Lattices , 1999, Inf. Process. Lett..

[11]  Lei Zhang,et al.  A Bottom-Up Incremental Algorithm of Building Concept Lattice , 2012 .

[12]  Derrick G. Kourie,et al.  AddIntent: A New Incremental Algorithm for Constructing Concept Lattices , 2004, ICFCA.

[13]  Gonzalo Urcid,et al.  Robust image retrieval from noisy inputs using lattice associative memories , 2009, Electronic Imaging.

[14]  Gerd Stumme,et al.  Efficient Mining of Association Rules Based on Formal Concept Analysis , 2005, Formal Concept Analysis.

[15]  Mondher Maddouri.  A Formal Concept Analysis Approach to Discover Association Rules from Data , 2005, CLA.

[16]  Vassilios Petridis,et al.  Fuzzy Lattice Neurocomputing (FLN) models , 2000, Neural Networks.

[17]  Vassilis G. Kaburlasos,et al.  Granular Enhancement of Fuzzy ART/SOM Neural Classifiers Based on Lattice Theory , 2007, Computational Intelligence Based on Lattice Theory.

[18]  Gerd Stumme,et al.  Conceptual knowledge discovery--a human-centered approach , 2003, Appl. Artif. Intell..

[19]  Gerhard X. Ritter,et al.  Neurons, Dendrites, and Pattern Classification , 2003, CIARP.

[20]  Yanwen Wu,et al.  Software Engineering and Knowledge Engineering: Theory and Practice , 2012 .

[21]  Ramakrishnan Srikant,et al.  Fast Algorithms for Mining Association Rules in Large Databases , 1994, VLDB.

[22]  Gerhard X. Ritter,et al.  Orthonormal Basis Lattice Neural Networks , 2006, FUZZ-IEEE.

[23]  Yang Huang,et al.  A Linear Delay Algorithm for Building Concept Lattices , 2008, CPM.

[24]  Rokia Missaoui,et al.  INCREMENTAL CONCEPT FORMATION ALGORITHMS BASED ON GALOIS (CONCEPT) LATTICES , 1995, Comput. Intell..

[25]  Sergei O. Kuznetsov,et al.  Comparing performance of algorithms for generating concept lattices , 2002, J. Exp. Theor. Artif. Intell..

[26]  Rokia Missaoui,et al.  Generating frequent itemsets incrementally: two novel approaches based on Galois lattice theory , 2002, J. Exp. Theor. Artif. Intell..

[27]  Oscar Camacho Nieto,et al.  An associative memory approach to medical decision support systems , 2012, Comput. Methods Programs Biomed..

[28]  Bernhard Ganter,et al.  Two Basic Algorithms in Concept Analysis , 2010, ICFCA.

[29]  Vilém Vychodil,et al.  Formal Concept Analysis With Background Knowledge: Attribute Priorities , 2009, IEEE Transactions on Systems, Man, and Cybernetics, Part C (Applications and Reviews).

[30]  Gerhard X. Ritter,et al.  Learning In Lattice Neural Networks that Employ Dendritic Computing , 2006, FUZZ-IEEE.