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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.… (More)

Kernel PCA as a nonlinear feature extractor has proven powerful as a preprocessing step for classification algorithms. But it can also be considered as a natural generalization of linear principal component analysis. This gives rise to the question how to use nonlinear features for data compression, reconstruction, and de-noising, applications common in… (More)

This paper provides an introduction to support vector machines, kernel Fisher discriminant analysis, and kernel principal component analysis, as examples for successful kernel-based learning methods. We first give a short background about Vapnik-Chervonenkis theory and kernel feature spaces and then proceed to kernel based learning in supervised and… (More)

We interpret several well-known algorithms for dimensionality reduction of manifolds as kernel methods. Isomap, graph Laplacian eigenmap, and locally linear embedding (LLE) all utilize local neighborhood information to construct a global embedding of the manifold. We show how all three algorithms can be described as kernel PCA on specially constructed Gram… (More)

We investigate a new kernel–based classifier: the Kernel Fisher Discrim-inant (KFD). A mathematical programming formulation based on the observation that KFD maximizes the average margin permits an interesting modification of the original KFD algorithm yielding the sparse KFD. We find that both, KFD and the proposed sparse KFD, can be understood in an… (More)

- Sebastian Mika, Stefan Jähnichen
- 2007

- Bernhard Schh, Sebastian Mika, Alex Smola, Gunnar Rr Atsch, Klaus-Robert M Uller
- 1998

Algorithms based on Mercer kernels construct their solutions in terms of expansions in a high-dimensional feature space F. Previous work has shown that all algorithms which can be formulated in terms of dot products in F can be performed using a kernel without explicitly working in F. The list of such algorithms includes support vector machines and… (More)

— We show via an equivalence of mathematical programs that a support vector (SV) algorithm can be translated into an equivalent boosting-like algorithm and vice versa. We exemplify this translation procedure for a new algorithm — one-class leveraging — starting from the one-class support vector machine (1-SVM). This is a first step towards un-supervised… (More)

Many settings of unsupervised learning can be viewed as quantization problems — the minimization of the expected quantization error subject to some restrictions. This allows the use of tools such as regularization from the theory of (supervised) risk minimization for un-supervised settings. Moreover, this setting is very closely related to both principal… (More)