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Splines with non positive kernels

2005

Conference Paper

ei


Non parametric regressions methods can be presented in two main clusters. The one of smoothing splines methods requiring positive kernels and the other one known as Nonparametric Kernel Regression allowing the use of non positive kernels such as the Epanechnikov kernel. We propose a generalization of the smoothing spline method to include kernels which are still symmetric but not positive semi definite (they are called indefinite). The general relationship between smoothing spline, Reproducing Kernel Hilbert Spaces and positive kernels no longer exists with indefinite kernel. Instead they are associated with functional spaces called Reproducing Kernel Krein Spaces (RKKS) embedded with an indefinite inner product and thus not directly associated with a norm. Smothing splines in RKKS have many of the interesting properties of splines in RKHS, such as orthogon ality, projection, representer theorem and generalization bounds. We show that smoothing splines can be defined in RKKS as the regularized solution of the interpolation problem. Since no norm is available in a RKKS, Tikhonov regularization cannot be defined. Instead, we proposed to use iterative methods of conjugate gradient type with early stopping as regularization mechanism. Several iterative algorithms were collected which can be used to solve the optimization problems associated with learning in indefinite spaces. Some preliminary experiments with indefinite kernels for spline smoothing are reported revealing the computational efficiency of the approach.

Author(s): Canu, S. and Ong, CS. and Mary, X.
Book Title: 5th International ISAAC Congress
Journal: More Progresses in Analysis: Proceedings of the 5th International ISAAC Congress
Pages: 1-10
Year: 2005
Month: July
Day: 0
Editors: Begehr, H. G.W., F. Nicolosi
Publisher: World Scientific

Department(s): Empirical Inference
Bibtex Type: Conference Paper (inproceedings)

Event Name: 5th International ISAAC Congress
Event Place: Catania, Italy

Address: Singapore
Digital: 0
Institution: International Society for Analysis, its Applications and Computation
Language: en
Organization: Max-Planck-Gesellschaft
School: Biologische Kybernetik

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BibTex

@inproceedings{3721,
  title = {Splines with non positive kernels},
  author = {Canu, S. and Ong, CS. and Mary, X.},
  journal = {More Progresses in Analysis: Proceedings of the 5th International ISAAC Congress},
  booktitle = {5th International ISAAC Congress},
  pages = {1-10},
  editors = {Begehr, H. G.W., F. Nicolosi},
  publisher = {World Scientific},
  organization = {Max-Planck-Gesellschaft},
  institution = {International Society for Analysis, its Applications and Computation},
  school = {Biologische Kybernetik},
  address = {Singapore},
  month = jul,
  year = {2005},
  month_numeric = {7}
}