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2013 | OriginalPaper | Chapter

7. Breaking Dimensions: Adaptive Scoring with Sparse Grids

Authors : Alexander Paprotny, Michael Thess

Published in: Realtime Data Mining

Publisher: Springer International Publishing

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Abstract

We introduce the concept of a sparse grid and show how this powerful approach to function space discretization may be employed to tackle high-dimensional machine learning problems of regression and classification. In particular, we address the issue of incremental computation of sparse grid regression coefficients so as to meet the requirements of realtime data mining. Conclusively, we present experimental results on real-world data sets.

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Literature
[ADT95]
go back to reference Arge, E., Daelen, M., Tveito, A.: Approximation of scattered data using smooth grid functions. J. Comput. Appl. Math. 59, 191–205 (1995)CrossRefMATHMathSciNet Arge, E., Daelen, M., Tveito, A.: Approximation of scattered data using smooth grid functions. J. Comput. Appl. Math. 59, 191–205 (1995)CrossRefMATHMathSciNet
[BGGK12]
go back to reference Bokanowski, O., Garcke, J., Griebel, M., Klompmaker, I.: An adaptive sparse grid semi-lagrangian scheme for first order Hamilton-Jacobi Bellman equations, submitted to J. Sci. Comput., also available as INS Preprint No. 1207 (2012) Bokanowski, O., Garcke, J., Griebel, M., Klompmaker, I.: An adaptive sparse grid semi-lagrangian scheme for first order Hamilton-Jacobi Bellman equations, submitted to J. Sci. Comput., also available as INS Preprint No. 1207 (2012)
[Bun92]
go back to reference Bungartz, H.-J: Dünne Gitter und deren Anwendung bei der adaptiven Lösung der dreidimensionalen Poisson-Gleichung (in German). Dissertation, TU München (1992) Bungartz, H.-J: Dünne Gitter und deren Anwendung bei der adaptiven Lösung der dreidimensionalen Poisson-Gleichung (in German). Dissertation, TU München (1992)
[BGRZ94]
go back to reference Bungartz, H.-J., Griebel, M., Röschke, D., Zenger, C.: Pointwise convergence of the combination technique for Laplace’s equation. East–west, J. Numer. Math. 2, 21–45 (1994)MATHMathSciNet Bungartz, H.-J., Griebel, M., Röschke, D., Zenger, C.: Pointwise convergence of the combination technique for Laplace’s equation. East–west, J. Numer. Math. 2, 21–45 (1994)MATHMathSciNet
[Diet98]
go back to reference Dietterich, T.G.: Approximate statistical tests for comparing supervised classification learning algorithms. Neural Comput. 10(7), 1895–1924 (1998)CrossRef Dietterich, T.G.: Approximate statistical tests for comparing supervised classification learning algorithms. Neural Comput. 10(7), 1895–1924 (1998)CrossRef
[EPP00]
[Gar06]
go back to reference Garcke J.: Regression with the optimized combination technique. In: ICML ’06: Proceedings of the 23th International Conference on Machine learning, pp. 321–328. ACM Press, New York (2006) Garcke J.: Regression with the optimized combination technique. In: ICML ’06: Proceedings of the 23th International Conference on Machine learning, pp. 321–328. ACM Press, New York (2006)
[FM01]
go back to reference Fung, G., Mangasarian, O.L.: Proximity support vector machine classifiers. In: Provost, F., Srikant, R. (eds.) Proceedings of the Seventh ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, pp. 77–86 (2001) Fung, G., Mangasarian, O.L.: Proximity support vector machine classifiers. In: Provost, F., Srikant, R. (eds.) Proceedings of the Seventh ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, pp. 77–86 (2001)
[Gar11]
go back to reference Garcke, J.: A dimension adaptive sparse grid combination technique for machine learning. In: Read, W., Larson, J.W., Roberts, A.J. (eds.) Proceedings of the 13th Biennial Computational Techniques and Application Conference, CTAC-2006, vol. 48 of ANZIAM J., pp. C725–C740 (2007) Garcke, J.: A dimension adaptive sparse grid combination technique for machine learning. In: Read, W., Larson, J.W., Roberts, A.J. (eds.) Proceedings of the 13th Biennial Computational Techniques and Application Conference, CTAC-2006, vol. 48 of ANZIAM J., pp. C725–C740 (2007)
[Gar12b]
go back to reference Garcke, J.: A dimension adaptive combination technique using localized adaptation criteria. In: Bock, H.G., Hoang, X.P., Rannacher, R., Schlöder, J.P. (eds.) Modeling, Simulation and Optimization of Complex Processes, pp. 115–125. Springer, Dordrecht (2012)CrossRef Garcke, J.: A dimension adaptive combination technique using localized adaptation criteria. In: Bock, H.G., Hoang, X.P., Rannacher, R., Schlöder, J.P. (eds.) Modeling, Simulation and Optimization of Complex Processes, pp. 115–125. Springer, Dordrecht (2012)CrossRef
[GG01a]
go back to reference Garcke, J., Griebel, M.: Data Mining with sparse grids using simplicial basis functions. In: Proceedings of the Seventh ACM SIGKDD International Conference on Knowledge Discovery and Data Mining (2001) Garcke, J., Griebel, M.: Data Mining with sparse grids using simplicial basis functions. In: Proceedings of the Seventh ACM SIGKDD International Conference on Knowledge Discovery and Data Mining (2001)
[Gir98]
go back to reference Girosi, F.: An equivalence between sparse approximation and support vector machines. Neural Comput. 10, 1455–1480 (1998)CrossRef Girosi, F.: An equivalence between sparse approximation and support vector machines. Neural Comput. 10, 1455–1480 (1998)CrossRef
[GJP93]
go back to reference Girosi, F., Jones, M., Poggio, T.: Priors, stabilizers and basis functions: from regularization to radial, tensor and additive splines. AI Memo No. 1430. Artificial Intelligence Laboratory, MIT (1993) Girosi, F., Jones, M., Poggio, T.: Priors, stabilizers and basis functions: from regularization to radial, tensor and additive splines. AI Memo No. 1430. Artificial Intelligence Laboratory, MIT (1993)
[GJP95]
go back to reference Girosi, F., Jones, M., Poggio, T.: Regularization theory and neural networks architectures. Neural Comput. 7, 219–265 (1995)CrossRef Girosi, F., Jones, M., Poggio, T.: Regularization theory and neural networks architectures. Neural Comput. 7, 219–265 (1995)CrossRef
[Gri92]
go back to reference Griebel, M.: The combination technique for the sparse grid solution of PDEs on multiprocessor machines. Parallel Process. Lett. 2(1), 61–70 (1992)CrossRefMathSciNet Griebel, M.: The combination technique for the sparse grid solution of PDEs on multiprocessor machines. Parallel Process. Lett. 2(1), 61–70 (1992)CrossRefMathSciNet
[GSZ92]
go back to reference Griebel, M., Schneider, M., Zenger, C.: A combination technique for the solution of sparse grid problems. In: de Groen, P., Beauwens, R. (eds.) Iterative Methods in Linear Algebra, pp. 263–281. Elsevier, Amsterdam (1992) Griebel, M., Schneider, M., Zenger, C.: A combination technique for the solution of sparse grid problems. In: de Groen, P., Beauwens, R. (eds.) Iterative Methods in Linear Algebra, pp. 263–281. Elsevier, Amsterdam (1992)
[Heg03]
go back to reference Hegland, M.: Adaptive sparse grids. In: Burrage, K., Sidje, R.B. (eds.) Proceedings of 10th Computational Techniques and Application Conference CTAC-2001, vol. 44, pp. C335–C353. (2003) Hegland, M.: Adaptive sparse grids. In: Burrage, K., Sidje, R.B. (eds.) Proceedings of 10th Computational Techniques and Application Conference CTAC-2001, vol. 44, pp. C335–C353. (2003)
[HL92]
go back to reference Hoschek, J., Lasser, D.: Grundlagen der geometrischen Datenverarbeitung. Teubner, Stuttgart (1992). Chapter 9 (in German)CrossRefMATH Hoschek, J., Lasser, D.: Grundlagen der geometrischen Datenverarbeitung. Teubner, Stuttgart (1992). Chapter 9 (in German)CrossRefMATH
[Kuh60]
go back to reference Kuhn, H.W.: Some combinatorial lemmas in topology. IBM J. Res. Develop. 4, 518–524 (1960)CrossRefMATH Kuhn, H.W.: Some combinatorial lemmas in topology. IBM J. Res. Develop. 4, 518–524 (1960)CrossRefMATH
[MM01]
go back to reference Mangasarian, O.L., Musicant, D.R.: Lagrangian Support Vector Machines, Journal of Machine Learning Research 1, 161-177 (2001) Mangasarian, O.L., Musicant, D.R.: Lagrangian Support Vector Machines, Journal of Machine Learning Research 1, 161-177 (2001)
[Pfl10]
go back to reference Pflüger D.: Spatial adaptive sparse grids for high-dimensional problems. Dissertation, TU München (2010) Pflüger D.: Spatial adaptive sparse grids for high-dimensional problems. Dissertation, TU München (2010)
[Rip94]
go back to reference Ripley, B.D.: Neural networks and related methods for classification. J. R. Stat. Soc. B 56(3), 409–456 (1994)MATHMathSciNet Ripley, B.D.: Neural networks and related methods for classification. J. R. Stat. Soc. B 56(3), 409–456 (1994)MATHMathSciNet
[Sal97]
go back to reference Salzberg, S.L.: On comparing classifiers: pitfalls to avoid and a recommended approach. Data Min. Knowl Discov. 1, 317–327 (1997)CrossRef Salzberg, S.L.: On comparing classifiers: pitfalls to avoid and a recommended approach. Data Min. Knowl Discov. 1, 317–327 (1997)CrossRef
[Sin98]
go back to reference Singh, S.: 2d spiral pattern recognition with possible measures. Pattern Recognit. Lett. 19(2), 141–147 (1998)CrossRefMATH Singh, S.: 2d spiral pattern recognition with possible measures. Pattern Recognit. Lett. 19(2), 141–147 (1998)CrossRefMATH
[Smo63]
go back to reference Smolyak, S.A.: Quadrature and interpolation formulas for tensor products of certain classes of functions. Dokl. Akad. Nauk SSSR. 148, 1042–1043 (1963). Russian, Engl. Transl.: Soviet Math. Dokl. 4, 240–243 (1963) Smolyak, S.A.: Quadrature and interpolation formulas for tensor products of certain classes of functions. Dokl. Akad. Nauk SSSR. 148, 1042–1043 (1963). Russian, Engl. Transl.: Soviet Math. Dokl. 4, 240–243 (1963)
[TA77]
go back to reference Tikhonov, A.N., Arsenin, V.A.: Solutions of Ill-Posed Problems. W.H. Winston, Washington, DC (1977)MATH Tikhonov, A.N., Arsenin, V.A.: Solutions of Ill-Posed Problems. W.H. Winston, Washington, DC (1977)MATH
[Wah90]
go back to reference Wahba, G.: Spline models for observational data, vol. 59. SIAM, Philadelphia (1990) Wahba, G.: Spline models for observational data, vol. 59. SIAM, Philadelphia (1990)
[Wie88]
go back to reference Wieland, A.: Spiral data set. CMU Artificial Intelligence repository, 1988 Wieland, A.: Spiral data set. CMU Artificial Intelligence repository, 1988
[Zen91]
go back to reference Zenger, C.: Sparse grids. In: Hackbusch, W. (ed.) Parallel Algorithms for Partial Differential Equations, Proceedings of the Sixth GAMM Seminar, Kiel, 1990, Vol. 31 of Notes on Num. Fluid Mech., pp. 241–251. Vieweg-Verlag (1991) Zenger, C.: Sparse grids. In: Hackbusch, W. (ed.) Parallel Algorithms for Partial Differential Equations, Proceedings of the Sixth GAMM Seminar, Kiel, 1990, Vol. 31 of Notes on Num. Fluid Mech., pp. 241–251. Vieweg-Verlag (1991)
Metadata
Title
Breaking Dimensions: Adaptive Scoring with Sparse Grids
Authors
Alexander Paprotny
Michael Thess
Copyright Year
2013
Publisher
Springer International Publishing
DOI
https://doi.org/10.1007/978-3-319-01321-3_7

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