ABSTRACT In this study, we visualize Pareto-optimum solutions derived from multiple-objective optimization using spherical self-organizing maps (SOMs) that lay out SOM data in three dimensions. There have been a wide range of studies involving plane SOMs where Pareto-optimal solutions are mapped to a plane. However, plane SOMs have an issue that similar data differing in a few specific variables are often placed at far ends of the map, compromising intuitiveness of the visualization. We show in this study that spherical SOMs allow us to find similarities in data otherwise undetectable with plane SOMs. We also implement and evaluate the performance using parallel sphere processing with several GPU environments.
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M. Yoshimi, T. Kuhara, K. Nishimoto, M. Miki and T. Hiroyasu, "Visualization of Pareto Solutions by Spherical Self-Organizing Map and It’s acceleration on a GPU," Journal of Software Engineering and Applications, Vol. 5 No. 3, 2012, pp. 129-137. doi: 10.4236/jsea.2012.53020.
 P. Czyzzak and A. Jaszkiewicz, “Pareto Simulated Annealing—A Metatheuristic Technique for Multiple-Objective Combinatorial Optimization,” Journal of Multi-Criteria Decision Analysis, Vol. 7, No. 7, 1998, 34-47.
 T. Kohonen, “The Self-Organizing Map,” Proceedings of the IEEE, Vol. 78, No.9, 1990, pp. 1464-1480.
 R. D. Prabhu, “SOMGPU: An Unsupervised Pattern Classifier on Graphical Processing Unit,” IEEE Congress on Evolutionary Computation, IEEE World Congress on Computational Intelligence, Hong Kong, 1-6 June 2008, pp. 1011-1018. doi: 10.1109/CEC.2008.4630920
 P. K. Kihato, H. Tokutaka, M. Ohkita, K. Fujimura, K. Kotani, Y. Kurozawa and Y. Maniwa, “Spherical and Torus SOM Approaches to Metabolic Syndrome Evaluation,” Neural Information Processing, Vol. 4985, 2008, pp. 274-284. doi:10.1007/978-3-540-69162-4_29
 Y. Wu and M. Takatsuka, “Spherical Self-Organizing Map Using Efficient Indexed Geodesic Data Structure,” Neural Networks, Vol. 19, No. 6-7, 2006, pp. 900-910.
 H. Tokutaka, P. K . Kihato, K. Fujimura and M. Ohkita, “Cluster Analysis using Spherical SOM,” Proceedings of the 6th International Workshop on Self-Organizing Maps. Bielefeld, 3-6 September 2007, pp. 1-7.
 G. A. Carpenter and S. Grossberg, “A Massively Parallel Architecture for a Self-Organizing Neural Pattern Recognition Machine,” Computer Vision, Graphics and Image Processing, Vol. 37, No. 1, 1987, pp. 54-115.
 H. Speckmann, P. Thole and W. Rosenstiel, “A COprocessor for KOhonen’s Self-Organizing Map (COKOS),” Proceedings of 1993 International Joint Conference on Neural Networks, Nagoya, 25-29 October 1993, pp. 1951-1954. doi:10.1109/IJCNN.1993.717038
 H. Tamukoh, T. Aso, K. Horio and T. Yamakawa, “Self-organizing Map Hardware Accelerator System and Its Application to Real Time Image Enlargement,” Proceedings of 2004 IEEE International Joint Conference on Neural Networks, Budapest, 25-29 July 2004, pp. 2686-2687.
 A. Shitara, Y. Nishikawa, M. Yoshimi and H. Amano, “Implementation and Evaluation of Self-Organizing Map Algorithm on a Graphic Processor,” Proceeding Parallel and Distributed Computing and Systems 2009, Cambridge, 2-4 November 2009.
 T. Hiroyasu, K. Kobayashi, M. Nishioka and M. Miki, “Diversity Maintenance Mechanism for Multi-Objective Genetic Algorithms Using Clustering and Network Inversion,” Lecture Notes in Computer Science, Vol. 5199, No. 1, 2008, pp. 722-732.
 S. Obayashi and D. Sasaki, “Visualization and Data Mining of Pareto Solutions Using Self-Organizing Map,” Proceedings of the 2nd International Conference on Evolutionary Multi-Criterion Optimization, Faro, 8-11 April, 2003, pp.796-809.