2 Kernels 1 2.1 Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2.2 Positive Definiteness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.3 General Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.4 Inner Product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.5 Dualit...

Transfinite barycentric kernels are the continuous version of traditional barycentric coordinates and are used to define interpolants of values given on a smooth planar contour. When the data is two-dimensional, i.e. the boundary of a planar map, these kernels may be conveniently expressed using complex number algebra, simplifying much of the notation and results. In this paper we develop some ...

SUMMARY The minimax kernels for nonparametric function and its derivative estimates are investigated. Our motivation comes from a study of minimax properties of nonparametric kernel estimates of probability densities and their derivatives. The asymptotic expression of the linear maximum risk is established. The corresponding minimax risk depends on the solutions to a kernel variational problem,...

Positive definite kernels, such as Gaussian Radial Basis Functions (GRBF), have been widely used in computer vision for designing feature extraction and classification algorithms. In many cases nonpositive definite (npd) kernels and non metric similarity/dissimilarity measures naturally arise (e.g., Hausdorff distance, Kullback Leibler Divergences and Compact Support (CS) Kernels). Hence, there...

In this paper, we propose a general smoothing framework for graph kernels by taking structural similarity into account, and apply it to derive smoothed variants of popular graph kernels. Our framework is inspired by state-of-the-art smoothing techniques used in natural language processing (NLP). However, unlike NLP applications that primarily deal with strings, we show how one can apply smoothi...

We propose a comprehensive survey of tree kernels through the lens of the mapping kernels framework. We argue that most existing tree kernels, as well as many more that are presented for the first time in this paper, fall into a typology of kernels whose seemingly intricate computation can be efficiently factorized to yield polynomial time algorithms. Despite this fact, we argue that a naive im...

Convolution kernels, constructed by convolution of sub-kernels defined on sub-structures of composite objects, are widely used in classification, where one important issue is to choose adequate sub-structures, particularly for objects such as trees, graphs, and sequences. In this paper, we study the problem of sub-structure selection for constructing convolution kernels by combining heterogeneo...

JKernelMachines is a Java library for learning with kernels. It is primarily designed to deal with custom kernels that are not easily found in standard libraries, such as kernels on structured data. These types of kernels are often used in computer vision or bioinformatics applications. We provide several kernels leading to state of the art classification performances in computer vision, as wel...