Random Projection RBF Nets for Multidimensional Density Estimation

Skubalska-Rafajłowicz, Ewa

doi:10.2478/v10006-008-0040-9

Abstract

The dimensionality and the amount of data that need to be processed when intensive data streams are observed grow rapidly together with the development of sensors arrays, CCD and CMOS cameras and other devices. The aim of this paper is to propose an approach to dimensionality reduction as a first stage of training RBF nets. As a vehicle for presenting the ideas, the problem of estimating multivariate probability densities is chosen. The linear projection method is briefly surveyed. Using random projections as the first (additional) layer, we are able to reduce the dimensionality of input data. Bounds on the accuracy of RBF nets equipped with a random projection layer in comparison to RBF nets without dimensionality reduction are established. Finally, the results of simulations concerning multidimensional density estimation are briefly reported.

References

Achlioptas D. (2001). Database friendly random projections, Proceedings of the 20th ACM SIGMOD-SIGACT-SIGART Symposium on Principles of Database Systems, Santa Barbara, CA, USA, pp. 274-281.
Search in Google Scholar
Ailon N. and Chazelle B. (2006). Approximate nearest neighbors and the fast Johnson-Lindenstrauss transform, Proceedings of the 38th Annual ACM Symposium on Theory of Computing, Seattle, WA, USA, pp. 557-563.
Search in Google Scholar
Arriaga R. and Vempala S. (1999). An algorithmic theory of learning: Robust concepts and random projection, Proceedings of the 40th Annual Symposium on Foundations of Computer Science, New York, NY, USA, pp. 616-623.
Search in Google Scholar
Bishop C. M. (1994). Novelty detection and neural-network validation, IEE Proceedings - Vision Image and Signal Processing, 141:217-222.10.1049/ip-vis:19941330
Search in Google Scholar
Bishop C. M. (1995). Neural Networks for Pattern Recognition, Oxford University Press, Oxford.10.1201/9781420050646.ptb6
Search in Google Scholar
Bowman A. W. (1995). An alternative method of cross-validation for the smoothing of density estimates, Biometrika 71(2): 353-360.10.1093/biomet/71.2.353
Search in Google Scholar
Broomhead D. and Lowe D. (1988). Multivariable functional interpolation and adaptive networks, Complex Systems 2(11): 321-323.
Search in Google Scholar
Buhmann M. D. (1988). Radial Basis Functions: Theory and Implementations, Cambridge University Press, Cambridge.
Search in Google Scholar
Chen S., Cowan C. F. N. and Grant P. M. (1991). Orthogonal least squares learning algorithm for radial basisfunction networks, IEEE Transactions on Neural Networks 2(2): 302-307.10.1109/72.80341
Search in Google Scholar
Chen S., Hong X. and Harris C. J. (2004). Sparse kernel density construction using orthogonal forward regression with leave-one-out test score and local regularization, IEEE Transactions on Systems, Man, and Cybernetics, Part B, 34(4): 1708-1717.10.1109/TSMCB.2004.828199
Search in Google Scholar
Dasgupta S. and Gupta A. (2003). An elementary proof of a theorem of Johnson and Lindenstrauss, Random Structures and Algorithms 22(1): 60-65.10.1002/rsa.10073
Search in Google Scholar
Devroye L. and Györfi L. (1985). Nonparametric Density Estimation. The L₁ View. Wiley, New York, NY.
Search in Google Scholar
Devroye L., Györfi L. and Lugosi G. (1996). Probabilistic Theory of Pattern Recognition, Springer-Verlag, New York, NY.10.1007/978-1-4612-0711-5
Search in Google Scholar
Frankl P. and Maehara H. (1987). The Johnson-Lindenstrauss lemma and the sphericity of some graphs, Journal of Combinatorial Theory A 44(3): 355-362.10.1016/0095-8956(88)90043-3
Search in Google Scholar
Gertler J. J. (1998). Fault Detection and Diagnosis in Engineering Systems, Marcel Dekker, New York, NY.
Search in Google Scholar
Guh R. (2005). A hybrid learning based model for on-line detection and analysis of control chart patterns, Computers and Industrial Engineering 49(1): 35-62.10.1016/j.cie.2005.03.002
Search in Google Scholar
Holmström L. and Hämäläinen A. (1993). The self-organizing reduced kernel density estimator, Proceedings of the 1993 IEEE International Conference on Neural Networks, San Francisco, CA, USA, 1: 417-421.
Search in Google Scholar
Haykin S. (1999). Neural Networks. A Comprehensive Foundation, 2nd Ed., Prentice-Hall, Upper Saddle River, NJ.
Search in Google Scholar
Indyk P. and Motwani R. (1998). Approximate nearest neighbors: Towards removing the curse of dimensionality, Proceedings of the 30th Annual ACM Symposium on Theory of Computing, Dallas, TX, USA, pp. 604-613.
Search in Google Scholar
Indyk P. and Naor A. (2006). Nearest neighbor preserving embeddings, ACM Transactions on Algorithms (to appear).10.1145/1273340.1273347
Search in Google Scholar
Johnson W. B. and Lindenstrauss J. (1984). Extensions of Lipshitz mapping into Hilbert space, Contemporary Mathematics 26: 189-206.10.1090/conm/026/737400
Search in Google Scholar
Jones M. C., Marron J. S. and Sheather S. J. (1996). A brief survey of bandwidth selection for density estimation, Journal of the American Statistical Association 91(433): 401-407.10.1080/01621459.1996.10476701
Search in Google Scholar
Karayiannis N. B. (1999). Reformulated radial basis neural networks trained by gradient descent, IEEE Transactions on Neural Networks 10(3): 657-671.10.1109/72.761725
Search in Google Scholar
Krzyżak A. (2001). Nonlinear function learning using optimal radial basis function networks, Journal on Nonlinear Analysis 47(1): 293-302.10.1016/S0362-546X(01)00177-8
Search in Google Scholar
Krzyżak A., Linder T. and Lugosi G. (2001). Nonparametric estimation and classification using radial basis function nets and empirical risk minimization, IEEE Transactions on Neural Networks 7(2): 475-487.10.1109/72.485681
Search in Google Scholar
Krzyżak A. and Niemann H. (2001). Convergence and rates of convergence of radial basis functions networks in function learning, Journal on Nonlinear Analysis 47(1): 281-292.10.1016/S0362-546X(01)00176-6
Search in Google Scholar
Krzyżak A. and Skubalska-Rafajłowicz E. (2004). Combining space-filling curves and radial basis function networks, Artificial Intelligence and Soft Computing ICAISC 2004. 7th International conference. Zakopane, Poland, Lecture Notes in Artificial Intelligence, 3070: 229-234, Springer-Verlag, Berlin.
Search in Google Scholar
Korbicz J., Kościelny J. M., Kowalczuk Z. and Cholewa W. (Eds) (2004). Fault diagnosis. Models, Artificial Intelligence, Applications, Springer-Verlag, Berlin.10.1007/978-3-642-18615-8
Search in Google Scholar
Leonard J. A., and Kramer M. A. (1990). Classifying process behaviour with neural networks: Strategies for improved training and generalization, Proceedings of the American Control Conference, San Diego, CA, USA, pp. 2478-2483.
Search in Google Scholar
Leonard J. A., and Kramer M. A. (1991). Radial basis networks for classifying process faults, IEEE Control Systems Magazine 11(3): 31-38.10.1109/37.75576
Search in Google Scholar
Li Y., Pont M. J. and Jones N. B. (2002). Improving the performance of radial basis function classifiers in condition monitoring and fault diagnosis applications where ‘unknown’ faults may occur, Pattern Recognition Letters 23(5): 569-577.10.1016/S0167-8655(01)00133-7
Search in Google Scholar
Li P., Hastie T. J. and Church K. W. (2007). Nonlinear Estimators and tail bounds for dimension reduction in 11 using Cauchy random projections, The Journal of Machine Learning Research 8(10): 2497-2532.
Search in Google Scholar
Li P., Hastie T. J. and Church K. W. (2006). Sub-Gaussian random projections, Technical report, Stanford University.
Search in Google Scholar
Magdon-Ismail M. and Atiya A. (2002). Density estimation and random variate generation using multilayer networks, IEEE Transactions on Neural Networks 13(3): 497-520.10.1109/TNN.2002.100012018244452
Search in Google Scholar
Moody J. and Darken C. J. (1989). Fast learning in networks of locally tuned processing units, Neural Computation 1(2): 281-294.10.1162/neco.1989.1.2.281
Search in Google Scholar
Patton R. J. (1994). Robust model-based fault diagnosis: The state of the art, Proceedings of the IFAC Symposium on Fault Detection Supervision and Safety of Technical Processes, Espoo, Finland, pp. 1-24.
Search in Google Scholar
Patton R. J., Chen J. and Benkhedda H. (2000). A study on neurofuzzy systems for fault diagnosis, International Journal of Systems Science 31(11): 1441-1448.10.1080/00207720050197811
Search in Google Scholar
Parzen E. (1962). On estimation of a probability density function and mode, Annals of Mathematical Statistics 33(3): 1065-1076.10.1214/aoms/1177704472
Search in Google Scholar
Poggio T. and Girosi F. (1990). Networks for approximation and learning, Proceedings of the IEEE 78(9): 484-1487.10.1109/5.58326
Search in Google Scholar
Powell M. J. D. (1987). Radial basis functions for multivariable interpolation: A review, in (J. C. Mason, M. G. Cox, Eds.) Algorithms for Approximation, Clarendon Press, Oxford, pp. 143-167.
Search in Google Scholar
Rafajłowicz E. (2006). RBF nets in fault localization, 8th International Conference on Artificial Intelligence and Soft Computing - ICAISC 2006. Zakopane, Poland, LNCS, Springer-Verlag, Berlin/Heidelberg, 4029/2006: 113-122.
Search in Google Scholar
Rafajłowicz E., Skubalska- Rafajłowicz E. (2003). RBF nets based on equidistributed points, Proceedings of 9th IEEE International Conference: Methods and Models in Automation and Robotics MMAR 2003, Szczecin, Poland, 2: 921-926.
Search in Google Scholar
Roberts S. (2000). Extreme value statistics for novelty detection in biomedical data processing, IEE Proceedings: Science, Measurement and Technology 147 (6): 363-367.10.1049/ip-smt:20000841
Search in Google Scholar
Schlorer H. and Hartman U. (1992). Mapping neural networks derived from the Parzen window estimator, Neural Networks 5(6): 903-909.10.1016/S0893-6080(05)80086-3
Search in Google Scholar
Skubalska-Rafajłowicz E. (2000). On using space-filling curves and vector quantization for constructing multidimensional control charts, Proceedings of the 5th on Conference Neural Network and Soft Computing, Zakopane, Poland, pp. 162-167.
Search in Google Scholar
Skubalska-Rafajłowicz E. (2006a). RBF neural network for probability density function estimation and detecting changes in multivariate processes, 8th International Conference: Artificial Intelligence and Soft Computing - ICAISC 2006. Zakopane, Poland, LNCS, Springer-Verlag, Berlin/Heidelberg 4029/2006: 133-141.10.1007/11785231_15
Search in Google Scholar
Skubalska-Rafajłowicz E. (2006b). Self-organizing RBF neural network for probability density function estimation, Proceedings of the 12th IEEE International Conference on Methods and Models in Automation and Robotics, Międzyzdroje, Poland, pp. 985-988.
Search in Google Scholar
Specht D. F. (1990). Probabilistic neural networks, Neural Networks 3(1): 109-118.10.1016/0893-6080(90)90049-Q
Search in Google Scholar
Vempala S. (2004). The Random Projection Method, American Mathematical Society, Providence, RI.10.1090/dimacs/065/01
Search in Google Scholar
Wettschereck D. and Dietterich T. (1992). Improving the performance of radial basis function networks by learning center locations, in (B. Spatz, Ed.) Advances in Neural Information Processing Systems, Morgan Kaufmann, San Mateo, CA, Vol. 4, pp. 1133-1140.
Search in Google Scholar
Willsky A. S. (1976). A survey of design methods for failure detection in dynamic systems, Automatica 12(6): 601-611.10.1016/0005-1098(76)90041-8
Search in Google Scholar
Xu L., Krzyżak A. and Yuille A. (1994). On radial basis function nets and kernel regression: Statistical consistency, convergence rates, and receptive field size, Neural Networks 7(4): 609-628.10.1016/0893-6080(94)90040-X
Search in Google Scholar
Yee P. V. and Haykin S. (2001). Regularized Radial Basis Function Networks: Theory and Applications, John Wiley, New York, NY.
Search in Google Scholar
Yin H. and Allinson N. M. (2001). Self-organising mixture networks for probability density estimation, IEEE Transactions on Neural Networks 12(2): 405-411.10.1109/72.914534
Search in Google Scholar
Zorriassatine F., Tannock J. D. T. and O'Brien C. (2003). Using novelty detection to identify abnormalities caused by mean shifts in bivariate processes, Computers and Industrial Engineering 44(3): 385-408.10.1016/S0360-8352(02)00215-2
Search in Google Scholar

Random Projection RBF Nets for Multidimensional Density Estimation

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