The responsibility weighted Mahalanobis kernel for semi-supervised training of support vector machines for classification

The responsibility weighted Mahalanobis (RWM) kernel considers structure information in data with help of a parametric density model.It is perfectly suited for semi-supervised learning as the parameters of the density model can be found in an unsupervised way.For semi-supervised learning the RWM kernel outperforms some other kernel functions including the Laplacian kernel (Laplacian SVM). SVM with RWM kernels can be parameterized as easily as an SVM with standard RBF kernels, as known heuristics for the RBF kernel can be transferred to the new kernel.Standard training techniques such as SMO and standard implementations of SVM such as LIBSVM can be used with the RWM kernel without any algorithmic adjustments or extensions.Results are shown for 20 publicly available benchmark data sets. 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 paper, 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.