Multiple kernel learning (MKL) partially solves the kernel selection problem in support vector machines and similar classifiers by minimizing the empirical risk over a subset of the linear combination of given kernel matrices. For large sample sets, the size of the kernel matrices becomes a numerical issue. In many cases, the kernel matrix is of low-efficient rank. However, the low-rank property is not efficiently utilized in MKL algorithms. Here, we suggest multiple spectral kernel learning that efficiently uses the low-rank property by finding a kernel matrix from a set of Gram matrices of a few eigenvectors from all given kernel matrices, called a spectral kernel set. We provide a new bound for the gaussian complexity of the proposed kernel set, which depends on both the geometry of the kernel set and the number of Gram matrices. This characterization of the complexity implies that in an MKL setting, adding more kernels may not monotonically increase the complexity, while previous bounds show otherwise.
Kernel methods are known to be effective for nonlinear multivariate analysis. One of the main issues in the practical use of kernel methods is the selection of kernel. There have been a lot of studies on kernel selection and kernel learning. Multiple kernel learning (MKL) is one of the promising kernel optimization approaches. Kernel methods are applied to various classifiers including Fisher discriminant analysis (FDA). FDA gives the Bayes optimal classification axis if the data distribution of each class in the feature space is a gaussian with a shared covariance structure. Based on this fact, an MKL framework based on the notion of gaussianity is proposed. As a concrete implementation, an empirical characteristic function is adopted to measure gaussianity in the feature space associated with a convex combination of kernel functions, and two MKL algorithms are derived. From experimental results on some data sets, we show that the proposed kernel learning followed by FDA offers strong classification power.