Input space versus feature space in kernel-based methods
Scholkopf, B.
Mika, S.
Burges, C.J.C.
Knirsch, P.
Muller, K.-R.
Ratsch, G.
Smola, A.J.
GMD FIRST, Berlin;
This paper appears in: Neural Networks, IEEE Transactions on
Publication Date: Sep 1999
Volume: 10,
Issue: 5
On page(s): 1000-1017
ISSN: 1045-9227
References Cited: 38
CODEN: ITNNEP
INSPEC Accession Number: 6362642
Digital Object Identifier: 10.1109/72.788641
Current Version Published: 2002-08-06
Abstract
This paper collects some ideas targeted at advancing our
understanding of the feature spaces associated with support vector (SV)
kernel functions. We first discuss the geometry of feature space. In
particular, we review what is known about the shape of the image of
input space under the feature space map, and how this influences the
capacity of SV methods. Following this, we describe how the metric
governing the intrinsic geometry of the mapped surface can be computed
in terms of the kernel, using the example of the class of inhomogeneous
polynomial kernels, which are often used in SV pattern recognition. We
then discuss the connection between feature space and input space by
dealing with the question of how one can, given some vector in feature
space, find a preimage (exact or approximate) in input space. We
describe algorithms to tackle this issue, and show their utility in two
applications of kernel methods. First, we use it to reduce the
computational complexity of SV decision functions; second, we combine it
with the kernel PCA algorithm, thereby constructing a nonlinear
statistical denoising technique which is shown to perform well on
real-world data
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