TY - JOUR
T1 - Input space versus feature space in kernel-based methods
AU - Schölkopf, Bernhard
AU - Mika, Sebastian
AU - Burges, Chris J.C.
AU - Knirsch, Philipp
AU - Müller, Klaus Robert
AU - Rätsch, Gunnar
AU - Smola, Alexander J.
N1 - Funding Information:
Manuscript received January 23, 1999; revised May 16, 1999. Part of this work was done while P. Knirsch was with Bell Labs and B. Schölkopf and A. J. Smola were with the Department of Engineering, Australian National University, Canberra. This work was supported by the ARC and the DFG under Grant Ja 379/52,71,91. B. Schölkopf was with GMD FIRST, 12489 Berlin, Germany. He is now with Mocrosoft Research Ltd., Cambridge CB2, U.K. S. Mika, K.-R. Müller, G. Rätsch, and A. J. Smola are with GMD FIRST, 12489 Berlin, Germany. C. J. C. Burges is with Bell Laboratories, Holmdel NJ, USA. P. Knirsch is with the Max-Planck-Institut für biologische Kybernetik, 72076 Tübingen, Germany. Publisher Item Identifier S 1045-9227(99)07268-9.
PY - 1999
Y1 - 1999
N2 - 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.
AB - 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.
UR - http://www.scopus.com/inward/record.url?scp=0032594954&partnerID=8YFLogxK
U2 - 10.1109/72.788641
DO - 10.1109/72.788641
M3 - Article
C2 - 18252603
AN - SCOPUS:0032594954
SN - 2162-237X
VL - 10
SP - 1000
EP - 1017
JO - IEEE Transactions on Neural Networks and Learning Systems
JF - IEEE Transactions on Neural Networks and Learning Systems
IS - 5
ER -