Kernels, regularization and differential equations

Pattern Recognition

Volumn 41, Issue 11, 2008, Pages 3271-3286

  Steinke, Florian ^a   Schölkopf, Bernhard ^a  

^a MAX PLANCK INSTITUTE FOR BIOLOGICAL CYBERNETICS   (Germany)

Author keywords

Differential equation; Gaussian process; Positive definite kernel; Reproducing kernel Hilbert space

Indexed keywords

ARTIFICIAL INTELLIGENCE; BANACH SPACES; BESSEL FUNCTIONS; CONTROL THEORY; DIFFERENTIAL EQUATIONS; DIFFERENTIATION (CALCULUS); DYNAMICAL SYSTEMS; EXTRAPOLATION; FINITE DIFFERENCE METHOD; GAUSSIAN DISTRIBUTION; GAUSSIAN NOISE (ELECTRONIC); HILBERT SPACES; INFERENCE ENGINES; LEARNING SYSTEMS; LINEAR EQUATIONS; MATHEMATICAL OPERATORS; MILITARY DATA PROCESSING; ORDINARY DIFFERENTIAL EQUATIONS; SCHRODINGER EQUATION; STANDARDS; STOCHASTIC MODELS; SUPPORT VECTOR MACHINES; TRELLIS CODES;

ELSEVIER (CO); FILTER METHODS; FINITE DIFFERENCE EQUATIONS; FINITE DOMAINS; GAUSSIAN PROCESS (GP); GENERAL (CO); IN ORDER; INFERENCE ALGORITHMS; KERNEL MACHINES; MACHINE LEARNING METHODS; MATHEMATICAL MACHINERY; POSITIVE DEFINITE KERNELS; REPRODUCING KERNEL-HILBERT SPACE (RKHS); STOCHASTIC DIFFERENTIAL EQUATION (SDE); VECTOR MACHINES;

DIFFERENCE EQUATIONS;

EID: 48149115201     PISSN: 00313203     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.patcog.2008.06.011     Document Type: Article

References (52)

1. Kienzle W., Bakir G., Franz M., and Schölkopf B. Face detection-efficient and rank deficient. Advances in Neural Information Processing Systems vol. 17 (2005), MIT Press, Cambridge, MA 673-680
2. Zhang L., Wu B., and Nevatia R. Detection and tracking of multiple humans with extensive pose articulation. Proceedings of the 2007 IEEE Computer Society Conference on Computer Vision and Pattern Recognition (2007) 1-8
3. Song X., Iordanescu G., and Wyrwicz A. One-class machine learning for brain activation detection. Proceedings of the 2007 IEEE Computer Society Conference on Computer Vision and Pattern Recognition (2007) 1-8
4. Sanguinetti G., Milo M., Rattray M., and Lawrence N. Accounting for probe-level noise in principal component analysis of microarray data. Bioinformatics 21 19 (2005) 3748-3754
5. Lawrence N.D., and Quiñonero-Candela J. Local distance preservation in the GP-LVM through back constraints. Proceedings of the International Conference in Machine Learning (2006) 513-520
6. Vapnik V.N. The Nature of Statistical Learning Theory (1995), Springer, New York
7. Burges C. A tutorial on support vector machines for pattern recognition. Data Min. Knowl. Discovery 2 2 (1998) 121-167
8. Schölkopf B., and Smola A. Learning with Kernels (2002), MIT Press, Cambridge, MA. http://www.learning-with-kernels.org URL 〈http://www.learning-with-kernels.org〉
9. Weston J., and Watkins C. Support vector machines for multiclass pattern recognition. Proceedings of the Seventh European Symposium on Artificial Neural Networks (1999)
10. Smola A., and Schölkopf B. A tutorial on support vector regression. Stat. Comput. 14 3 (2004) 199-222
11. Schölkopf B., Platt J., Shawe-Taylor J., Smola A., and Williamson R. Estimating the support of a high-dimensional distribution. Neural Comput. 13 7 (2001) 1443-1471
12. Chapelle O., Schölkopf B., and Zien A. Semi-supervised Learning (2006), MIT Press, Cambridge, MA
13. Schölkopf B., Smola A., and Müller K. Nonlinear component analysis as a kernel eigenvalue problem. Neural Comput. 10 5 (1998) 1299-1319
14. Williams C., and Barber D. Bayesian classification with Gaussian processes. IEEE Trans. Pattern Anal. Mach. Intell. 20 12 (1998) 1342-1351
15. Opper M., and Winther O. Gaussian processes for classification: mean-field algorithms. Neural Comput. 12 11 (2000) 2655-2684
16. Williams C., and Rasmussen C. Gaussian processes for regression. Advances in Neural Information Processing Systems vol. 8 (1996) 514-520
17. Lawrence N., Seeger M., and Herbrich R. Fast sparse Gaussian process methods: the informative vector machine. Advances in Neural Information Processing Systems vol. 15 (2003) 609-616
18. Lawrence N. Gaussian process latent variable models for visualisation of high dimensional data. Advances in Neural Information Processing Systems vol. 16 (2004)
19. Smola A., Schölkopf B., and Müller K. The connection between regularization operators and support vector kernels. Neural Networks 11 4 (1998) 637-649
20. Girosi F., Jones M., and Poggio T. Priors stabilizers and basis functions: from regularization to radial, tensor and additive splines, A.I. Memo No. 1430 (1993), MIT Press, Cambridge, MA
21. Bogachev V. Gaussian Measures (1998), AMS, New York
22. Oksendal B. Stochastic differential equations: an introduction with applications. sixth ed. (2002), Springer, Berlin
23. Rasmussen C.E., and Williams C.K. Gaussian Processes for Machine Learning (2006), MIT Press, Cambridge, MA
24. Wahba G. Spline models for observational data (1990), SIAM, Philadelphia, PA
25. Ramsay J.O., and Silverman B.W. Functional Data Analysis. second ed. (2005), Springer, Berlin
26. Wendland H. Scattered Data Approximation (2005), Cambridge University Press, Cambridge, UK
27. Ljung L. System Identification-Theory for the User. second ed. (1999), Prentice-Hall, Upper Saddle River, NJ
28. Kimeldorf G., and Wahba G. A correspondence between bayesian estimation on stochastic processes and smoothing by splines. Ann. Math. Stat. 41 2 (1970) 495-502
29. Madych W., and Nelson S. Multivariate interpolation and conditionally positive definite functions. II. Math. Comput. 54 189 (1990) 211-230
30. Smola A., Schölkopf B., and Müller K.-R. The connection between regularization operators and support vector kernels. Neural Networks 11 (1998) 637-649
31. T. Graepel, Solving noisy linear operator equations by Gaussian processes: application to ordinary and partial differential equations, in: Proceedings of the 20th International Conference on Machine Learning, vol. 20, 2003, pp. 234-241.
32. M. Hein, O. Bousquet, Kernels, associated structures and generalizations, Technical Report 127, Max Planck Institute for Biological Cybernetics, Tübingen, Germany, 2004.
33. Heckman N.E., and Ramsay J.O. Penalized regression with model-based penalties. Can. J. Stat. 28 (2000) 241-258
34. Steinke F., and Schölkopf B. Machine learning methods for estimating operator equations. Proceedings of the 14th IFAC Symposium on System Identification, SYSID06 (2006), Elsevier, Amsterdam 1-6
35. Girosi F., Jones M., and Poggio T. Regularization theory and neural network architectures. Neural Comput. 7 (1995) 219-267
36. Walder C., Schölkopf B., and Chapelle O. Implicit surface modelling with a globally regularised basis of compact support. Comput. Graphics Forum 25 3 (2006) 635-644
37. Micchelli C., and Pontil M. On learning vector-valued functions. Neural Comput. 17 1 (2005) 177-204
38. Curtain R., and Zwart H. An Introduction to Infinite Dimensional Linear Systems Theory (1995), Springer, Berlin
39. Bochner S. Monotone Funktionen, Stieltjessche integrate und harmonische analyse. Math. Ann. 108 (1933) 378-410
40. Kalman R. A new approach to linear filtering and prediction problems. J. Basic Eng. 82 1 (1960) 35-45
41. Floater M., and Hormann K. Surface parameterization: a tutorial and survey. Advances in Multiresolution for Geometric Modelling vol. 1 (2005), Springer, Berlin
42. Hein M., Audibert J.-Y., and von Luxburg U. Graph Laplacians and their convergence on random neighborhood graphs. J. Mach. Learn. Res. 8 (2007) 1325-1370
43. Smola A., and Kondor R. Kernels and regularization on graphs. Proceedings of the Conference on Learning Theory (2003), Springer, Berlin
44. X. Zhu, Z. Ghahramani, J. Lafferty, Semi-supervised learning using Gaussian fields and harmonic functions, in: Proceedings of the 20th International Conference on Machine Learning, vol. 20, 2003.
45. M. Tipping, C. Bishop, Bayesian image super-resolution, in: Advances in Neural Information Processing Systems, vol. 15, 2003, pp. 1279-1286.
46. von Luxburg U. A tutorial on spectral clustering. Stat. Comput. 17 4 (2007) 395-416
47. Jordan M., Ghahramani Z., Jaakkola T., and Saul L. An introduction to variational methods for graphical models. Mach. Learning 37 2 (1999) 183-233
48. Huang Y., and McColl W. Analytical inversion of general tridiagonal matrices. J. Phys. A: Math. General 30 (1997) 7919-7933
49. C. Archambeau, D. Cornford, M. Opper, J. Shawe-Taylor, Gaussian process approximations of stochastic differential equations, J. Mach. Learn. Res., in: Workshop and Conference Proceedings, vol. 1, 2007, pp. 1-16.
50. S. Julier, J. Uhlmann, A new extension of the Kalman filter to nonlinear systems, in: I. Kadar (Ed.), Proceedings of the Conference on Signal Processing, Sensor Fusion, and Target Recognition VI, vol. 3068, 1997, pp. 182-193.
51. Doucet A., de Freitas N., and Gordon N. Sequential Monte Carlo Methods in Practice (2001), Springer, Berlin
52. H. Heuser, Lehrbuch der Analysis, Teil 2, B. G. Teubner, Stuttgart, Germany, 1991.