Abstract:
We introduce a new method of constructing kernels on sets whose elements are discrete structures like strings, trees and graphs. The method can be applied iteratively to build a kernel on a infinite set from kernels involving generators of the set. The family of kernels generated generalizes the family of radial basis kernels. It can also be used to define kernels in the form of joint Gibbs probability distributions. Kernels can be built from hidden Markov random fields, generalized regular expressions, pair-HMMs, or ANOVA decompositions. Uses of the method lead to open problems involving the theory of infinitely divisible positive definite functions. Fundamentals of this theory and the theory of reproducing kernel Hilbert spaces are reviewed and applied in establishing the validity of the method. 1 Introduction Many problems in statistics and pattern recognition demand that discrete structures likes strings, trees, and graphs be classified or clustered based on similarity. To do th...
Citations
|
5044
|
Statistical Learning Theory
– Vapnik
- 1998
|
|
2870
|
Introduction to automata theory, languages and computation
– Hopcroft, Ullman
- 1979
|
|
2439
|
Stochastic Relaxation, Gibbs Distributions, and the Bayesian Restoration of Images
– Geman, Geman
- 1984
|
|
2372
|
A tutorial on hidden Markov Models and selected applications in speech recognition
– Rabiner
- 1989
|
|
1240
|
A tutorial on support vector machines for pattern recognition
– Burges
- 1998
|
|
1040
|
An Introduction to Probability Theory and Its Applications, Volume I, 3rd Edition
– Feller
- 1968
|
|
769
|
Spline Models for Observational Data
– WAHBA
- 1990
|
|
588
|
Graphical Models
– Lauritzen
- 1996
|
|
457
|
Linear Algebra and Its Applications
– Strang
- 1976
|
|
441
|
Biological sequence analysis—- Probabilistic models of proteins and nucleic acids. Combridge
– Durbin, Eddy, et al.
- 1998
|
|
320
|
The Geometry of Graphs and Some of Its Algorithmic Applications
– Linial, London, et al.
- 1995
|
|
298
|
Time Warps, String Edits and Macromolecules: the Theory and Practice of Sequence Comparisons
– Sankoff, Kruskal
- 1983
|
|
264
|
Exploiting generative models in discriminative classifiers
– Jaakkola, Haussler
- 1998
|
|
170
|
Syntactic Pattern Recognition and Applications
– Fu
- 1982
|
|
163
|
Real Analysis and Probability
– Dudley
- 1989
|
|
143
|
An equivalence between sparse approximation and support vector machines
– Girosi
- 1998
|
|
102
|
Support vector machines, reproducing kernel Hilbert spaces, and randomized gacv
– Wahba
- 1998
|
|
100
|
Using the fisher kernel method to detect remote protein homologies
– Jaakkola, Diekhans, et al.
- 1999
|
|
89
|
Probabilistic kernel regression models
– Jaakkola, Haussler
- 1999
|
|
86
|
Comparing support vector machines with gaussian kernels to radial basis function classifiers
– Scholkopf, Sung, et al.
- 1996
|
|
64
|
Introduction to Gaussian processes
– MacKay
- 1998
|
|
62
|
Ridge regression learning algorithm in dual variables
– Saunders, Gammerman, et al.
- 1998
|
|
56
|
Harmonic Analysis on Semigroups (Theory of Positive Definite
– Berg, Christensen, et al.
- 1984
|
|
41
|
Grammatical inference: Introduction and survey -- part 2
– Fu, Booth
- 1975
|
|
34
|
A sparse representation for function approximation
– Poggio, Girosi
- 1998
|
|
33
|
Global self-organization of all known protein sequences reveals inherent biological signatures
– Linial, Linial, et al.
- 1997
|
|
6
|
On fractional hadamard powers of positive definite matrices
– FitzGerald, Horn
- 1977
|
|
2
|
Hilbert Space Methods
– M'at'e
- 1989
|
|
1
|
Positive powers of positive positive definite matrices
– Rosen
- 1996
|
