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7 th Workshop on Intelligent and Knowledge Oriented Technologies Smolenice 22-23.11.2012 WIKT 2012 Reduction of Computation Times of GOSCL Algorithm Using Sparse-based Implementation Peter Butka Department of Cybernetics and Artificial Intelligence Faculty of Electrical Engineering and Informatics Technical University of Košice Slovakia Email: peter.butka@tuke.skpeter.butka@tuke.sk
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Reduction of Computation Times of GOSCL Algorithm Using Sparse-based Implementation WIKT 2012 Formal Concept Analysis (FCA) Basics of FCA Conceptual data analysis method, crisp case introduces by Ganter&Wille, based on Galois connections, output: concept lattice B - objects, A - attributes, I B A – relation, then there are mappings and, for which: Concept lattice: One-sided Concept Lattices Krajči, Yahia&Jaoua, one “side” of context is fuzzified (attributes) Input: L-context (B,A,R), binary L-fuzzy relation R: B A L, L is a complete lattice. Then we have and, for which: Concept lattice is defined in same way as in classical FCA case (mappings , form Galois connection which induces concept lattice) 2
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Reduction of Computation Times of GOSCL Algorithm Using Sparse-based Implementation WIKT 2012 Theory of GOSCL Generalized one-sided formal context B – non-empty set of objects, A – non-empty set of attributes , CL – class of all complete lattices For a A denotes truth value structure for attribute a Generalized incidence relation R – R(b,a) Generalized one-sided concept lattice Mappings: and These mappings form concept lattice between and for all pairs,, X – extent, g - intent satisfying and, with partial order defined as: Advantage: works for any type of attribute (nominal, ordinal, real,…) 3
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Reduction of Computation Times of GOSCL Algorithm Using Sparse-based Implementation WIKT 2012 Algorithm (GOSCL) Incremental algorithm for context R(b)(a)= R(b)(a), i.e. R(b) is b-th row of data table represented by R denote greatest element of, i.e. 4
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Reduction of Computation Times of GOSCL Algorithm Using Sparse-based Implementation WIKT 2012 Example 5
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Reduction of Computation Times of GOSCL Algorithm Using Sparse-based Implementation WIKT 2012 Complexity of GOSCL for Sparse Data Tables Sparse-based implementation Representation of sparse matrices (BLAS impl. in JBOWL library) Sparse implementation of meet operation Sparse implementation of operation Experiments Preparing of data for experiments Interval-based attributes of reals from [0,1], generated randomly to the real frequencies of ‘zeros’ of text-mining dataset (Reuters) Different sparseness (s {0.0, 0.1,..., 0.9}) - indicates num. of ‘zeros’ Effect of sparseness on time complexity (and size of lattice) for fixed number of objects with different implementations increasing number of inputs (with fixed s) and different implementations fixed s and number of objects – number of attributes are changing for different implementations for fixed sparseness (Reuters) and attributes – changing number of objects, for different implementations + reduction ratio ST/SP2 implem. 6
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Reduction of Computation Times of GOSCL Algorithm Using Sparse-based Implementation WIKT 2012 Results of experiments 7
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Reduction of Computation Times of GOSCL Algorithm Using Sparse-based Implementation WIKT 2012 Conclusions and Future Work Conclusions Experimental analysis of influence of sparseness on time (and size) complexity GOSCL algorithm for standard and specialized sparse- based implementation It was shown that complexity is even more reduced with the increase of the sparseness of object-attribute model if sparse-based implementation is used It is important for domains with large sparse data tables Goal: to analyze ‘large’ contexts Usage of large models in retrieval using projections from large lattice Future work Experiments with real data examples of sparse domains (e.g. direct usage of text-mining datasets, etc.) Reuse of implementation in distributed version of GOSCL for grid/cloud implementations and their testing 8
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