  Kernel functions in support vector machines (SVM) are needed to assess the
similarity of input samples in order to classify these samples, for instance.
Besides standard kernels such as Gaussian (i.e., radial basis function, RBF) or
polynomial kernels, there are also specific kernels tailored to consider
structure in the data for similarity assessment. In this article, we will
capture structure in data by means of probabilistic mixture density models, for
example Gaussian mixtures in the case of real-valued input spaces. From the
distance measures that are inherently contained in these models, e.g.,
Mahalanobis distances in the case of Gaussian mixtures, we derive a new kernel,
the responsibility weighted Mahalanobis (RWM) kernel. Basically, this kernel
emphasizes the influence of model components from which any two samples that
are compared are assumed to originate (that is, the "responsible" model
components). We will see that this kernel outperforms the RBF kernel and other
kernels capturing structure in data (such as the LAP kernel in Laplacian SVM)
in many applications where partially labeled data are available, i.e., for
semi-supervised training of SVM. Other key advantages are that the RWM kernel
can easily be used with standard SVM implementations and training algorithms
such as sequential minimal optimization, and heuristics known for the
parametrization of RBF kernels in a C-SVM can easily be transferred to this new
kernel. Properties of the RWM kernel are demonstrated with 20 benchmark data
sets and an increasing percentage of labeled samples in the training data.
