Classification with sparse grids using simplicial basis functions

Intelligent Data Analysis

Volumn 6, Issue 6, 2002, Pages 483-502

  Garcke, Jochen ^a   Griebel, Michael ^a  

^a UNIVERSITY OF BONN   (Germany)

Author keywords

approximation; classification; combination technique; data mining; simplicial discretization; sparse grids

Indexed keywords

COMPUTATIONAL COMPLEXITY; DATA MINING; FUNCTIONS; MESH GENERATION;

ANISOTROPIC SPARSE GRIDS; APPROXIMATION; COMBINATION TECHNIQUE; CURSE OF DIMENSIONALITY; DATA MINING APPLICATIONS; DISCRETIZATIONS; HIGH-DIMENSIONAL FEATURE SPACE; SPARSE GRID;

CLASSIFICATION (OF INFORMATION);

EID: 0141944503     PISSN: 1088467X     EISSN: 15714128     Source Type: Journal    
DOI: 10.3233/ida-2002-6602     Document Type: Article

References (52)

1. E. Arge, M. Dæhlen and A. Tveito, Approximation of scattered data using smooth grid functions, J. Comput. Appl. Math 59 (1995), 191-205.
2. R. Balder, Adaptive Verfahren f̈ur elliptische und parabolische Differentialgleichungen auf d̈unnen Gittern, Dissertation, Technische Universiẗat M̈unchen, 1994.
3. G. Baszenski, N-th order polynomial spline blending, in: Multivariate Approximation Theory III, W. Schempp and K. Zeller, eds, ISNM 75, Birkḧauser, Basel, 1985, pp. 35-46.
4. S.D. Bay, The UCI KDD archive, http://kdd.ics.uci.edu, 1999.
5. M.J.A. Berry and G.S. Linoff, Mastering Data Mining, Wiley, 2000.
6. C.L. Blake and C.J. Merz, UCI repository of machine learning databases, 1998, http://www.ics.uci.edu/̃mlearn/MLRepository. html.
7. D. Botkin, J. Janak and J. Wallis, Some ecological conseque nces of a computer model of forest growth, J. Ecology 60 (1972), 849-872.
8. H.-J. Bungartz, D̈unne Gitter und deren Anwendung bei der adaptiven L̈osung der dreidimensionalen Poisson-Gleichung, Dissertation, Institut f̈ur Informatik, Technische Universiẗat M̈unchen, 1992.
9. H.-J. Bungartz, T. Dornseifer and C. Zenger, Tensor product approximation spaces for the efficient numerical solution of partial differential equations, in: Proc. Int.Workshop on Scientific Computations, Konya, 1996, Nova Science Publishers, to appear, 1997.
10. H.-J. Bungartz, M. Griebel, D. R̈oschke and C. Zenger, Pointwise convergence of the combination technique for the Laplace equation, East-West J. Numer. Math. 2 (1994), 21-45.
11. V. Cherkassky and F. Mulier, Learning from Data-Concepts, Theory and Methods, John Wiley and Sons, 1998.
12. K. Cios, W. Pedrycz R. Swiniarski, Data Mining Methods for Knowledge Discovery, Kluwer, 1998.
13. T.G. Dietterich, Approximate statistical tests for comparing supervised classification learning algorithms, Neural Computation 10(7) (1998), 1895-1924.
14. T. Evgeniou, M. Pontil and T. Poggio, Regularization networks and support vector machines, Advances in Computational Mathematics 13 (2000), 1-50.
15. K. Frank, S. Heinrich and S. Pereverzev, Information Complexity of Multivariate Fredholm Integral Equations in Sobolev Classes, J. of Complexity 12 (1996), 17-34.
16. H. Freudenthal, Simplizialzerlegungen von beschr̈ankter Flachheit, Annals of Mathematics 43 (1942), 580-582.
17. G. Fung and O. Mangasarian, Proximal support vector machine classifiers, in: Proceedings of the Seventh ACMSIGKDD International Conference on Knowledge Discovery and Data Mining, F. Provost and R. Srikant, eds, 2001, pp. 77-86.
18. J. Garcke and M. Griebel, On the computation of the eigenproblems of hydrogen and helium in strong magnetic and electric fields with the sparse grid combination technique, Journal of Computational Physics 165(2) (2000), 694-716.
19. J. Garcke and M. Griebel, On the parallelization of the sparse grid approach for data mining, in: Large-Scale Scientific Computations, Third International Conference, LSSC 2001, Sozopol, Bulgaria, vol. 2179 of Lecture Notes in Computer Science, 2001, pp. 22-32.
20. J. Garcke, M. Griebel M. Thess, Data mining with sparse grids, Computing 67(3) (2001), 225-253.
21. T. Gerstner and M. Griebel, Numerical Integration using Sparse Grids, Numer. Algorithms 18 (1998), 209-232.
22. F. Girosi, An equivalence between sparse approximation and support vector machines, Neural Computation 10(6) (1998), 1455-1480.
23. F. Girosi, M. Jones and T. Poggio, Regularization theory and neural networks architectures, Neural Computation 7 (1995), 219-265.
24. G. Golub, M. Heath G. Wahba, Generalized cross validation as a method for choosing a good ridge parameter, Technometrics 21 (1979), 215-224.
25. M. Griebel, The combination technique for the sparse grid solution of PDEs on multiprocessor machines, Parallel Processing Letters 2(1) (1992), 61-70.
26. M. Griebel, Adaptive sparse grid multilevel methods for elliptic PDEs based on finite differences, Computing 61(2) (1998), 151-179.
27. M. Griebel,W. Huber, T. St ̈ortkuhl and C. Zenger, On the parallel solution of 3D PDEs on a network of workstations and on vector computers, in: Parallel Computer Architectures: Theory, Hardware, Software, Applications, A. Bode and M. D. Cin, eds, vol. 732 of Lecture Notes in Computer Science, Springer Verlag, 1993, pp. 276-291.
28. M. Griebel and S. Knapek, Optimized tensor-product approximation spaces, Constructive Approximation 16(4) (2000), 525-540.
29. M. Griebel, P. Oswald T. Schiekofer, Sparse grids for boundary integral equations, Numer. Mathematik 83(2) (1999), 279-312.
30. M. Griebel, M. Schneider C. Zenger. A combination technique for the solution of sparse grid problems. in: Iterative Methods in Linear Algebra, P. de Groen and R. Beauwens, eds, IMACS, Elsevier, North Holland, 1992, pp. 263-281.
31. M. Griebel and V. Thurner, The efficient solution of fluid dynamics problems by the combination technique, Int. J. Num. Meth. for Heat and Fluid Flow 5(3) (1995), 251-269.
32. M. Hegland, O.M. Nielsen and Z. Shen, High dimensional smoothing based on multilevel analysis, Technical report, Data Mining Group, The Australian National University, Canberra, November 2000, Submitted to SIAM J. Scientific Computing.
33. J. Hoschek and D. Lasser, Grundlagen der goemetrischen Datenverarbeitung, chapter 9, Teubner, 1992.
34. H.W. Kuhn, Some combinatorial lemmas in topology, IBM J. Res. Develop. 4 (1960), 518-524.
35. O.L. Mangasarian and D.R. Musicant, Lagrangian support vector machines, Journal of Machine Learning Research 1 (2001), 161-177.
36. G. Melli, Datgen: A program that creates structured data, Website, http://www.datasetgenerator.com.
37. S. Odewahn, E. Stockwell, R. Pennington, R. Humphreys, W. Zumach, Automated star/galaxy discrimination with neural networks, Astronomical Journal 103(1) (1992), 318-331.
38. W.D. Penny and S.J. Roberts, Bayesian neural networks for classification: how useful is the evidence framework? Neural Networks 12 (1999), 877-892.
39. B.D. Ripley, Neural networks and related methods for classification, Journal of the Royal Statistical Society B 56(3) (1994), 409-456.
40. S.L. Salzberg, On comparing classifiers: Pitfalls to avoid and a recommended approach, Data Mining and Knowledge Discovery 1 (1997), 317-327.
41. T. Schiekofer, Die Methode der Finiten Differenzen auf d̈unnen Gittern zur L̈osung elliptischer und parabolischer partieller Differentialgleichungen, Dissertation, Institut f̈ur Angewandte Mathematik, Universiẗat Bonn, 1999.
42. W. Sickel and F. Sprengel, Interpolation on sparse grids and Nikol'skij-Besov spaces of dominating mixed smoothness, J. Comput. Anal. Appl. 1 (1999), 263-288.
43. S. Singh, 2d spiral pattern recognition with possibilistic measures, Pattern Recognition Letters 19(2) (1998), 141-147.
44. S.A. Smolyak, Quadrature and interpolation formulas for tensor products of certain classes of functions, Dokl. Akad. Nauk SSSR 148 (1963), 1042-1043. Russian, Engl. Transl.: Soviet Math. Dokl. 4 (1963), 240-243.
45. V.N. Temlyakov, Approximation of functions with bounded mixed derivative, Proc. Steklov Inst. Math. 1 (1989).
46. A.N. Tikhonov and V.A. Arsenin, Solutions of ill-posed problems, W.H. Winston, Washington D.C., 1977.
47. F. Utreras, Cross-validation techniques for smoothing spline functions in one or two dimensions, in: Smoothing techniques for curve estimation, T. Gasser and M. Rosenblatt, eds, Springer-Verlag, Heidelberg, 1979, pp. 196-231.
48. V.N. Vapnik, Estimation of dependences based on empirical data, Springer-Verlag, Berlin, 1982.
49. V.N. Vapnik, The Nature of Statistical Learning Theory, Springer, 1995.
50. G. Wahba, Spline models for observational data, volume 59 of Series in Applied Mathematics, SIAM, Philadelphia, 1990.
51. A. Wieland, Spiral data set, http://www.cs.cmu.edu/afs/cs.cmu.edu/ project/ai-repository/ai/areas/neural/b0.html.
52. C. Zenger, Sparse grids, in: Parallel Algorithms for Partial Differential Equations, Proceedings of the Sixth GAMMSeminar, Kiel, 1990, W. Hackbusch, ed., volume 31 of Notes on Num. Fluid Mech. Vieweg-Verlag, 1991.