1. 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

2. 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

3. 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

4. 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

5. Reduction of Computation Times of GOSCL Algorithm Using Sparse-based Implementation WIKT 2012 Example 5

6. 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

7. Reduction of Computation Times of GOSCL Algorithm Using Sparse-based Implementation WIKT 2012 Results of experiments 7

8. 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