Input space versus feature space in kernel-based methods

TLDR

The paper seeks to advance understanding of support vector kernel feature spaces and to develop algorithms for locating preimages of feature vectors in input space. The authors examine the geometry of feature space, compute its intrinsic metric from the kernel (illustrated with inhomogeneous polynomial kernels), and outline methods for mapping feature vectors back to input space. The proposed algorithms reduce the computational complexity of SV decision functions and, when combined with kernel PCA, yield a nonlinear denoising technique that performs well on real‑world data.

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.