We characterize the reproducing kernel Hilbert spaces whose elements are p-integrable functions in terms of the boundedness of the integral operator whose kernel is the reproducing kernel. Moreover, for p = 2 we show that the spectral decomposition of this integral operator gives a complete description of the reproducing kernel.

Let H m be the reproducing kernel Hilbert space with the kernel function (z, w) ∈ B×B → (1− m ∑ i=1 ziw̄i) . We show that if θ : B → L(E , E∗) is a multiplier for which the corresponding multiplication operator Mθ ∈ L(H m ⊗ E , H 2 m ⊗ E∗) has closed range, then the quotient module Hθ, given by · · · −→ H m ⊗ E Mθ −→ H m ⊗ E∗ πθ −→ Hθ −→ 0, is similar to H m ⊗F for some Hilbert space F if and on...

Tree structured graphical models are powerful at expressing long range or hierarchical dependency among many variables, and have been widely applied in different areas of computer science and statistics. However, existing methods for parameter estimation, inference, and structure learning mainly rely on the Gaussian or discrete assumptions, which are restrictive under many applications. In this...

Due to its promising classification performance, sparse representation based classification(SRC) algorithm has attracted great attention in the past few years. However, the existing SRC type methods apply only to vector data in Euclidean space. As such, there is still no satisfactory approach to conduct classification task for symmetric positive definite (SPD) matrices which is very useful in c...

We show that polynomials do not belong to the reproducing kernel Hilbert space of infinitely differentiable translation-invariant kernels whose spectral measures have moments corresponding a det...

PCA in a reproducing kernel Hilbert space is analysed as probabilistically optimal procedure of dimension reduction given a covariance structure by the reproducing kernel. It provides a unifying framework for various seemingly disparate and special techniques of dimension reduction applied to splines, in geostatistical “kriging” or in interpolation of data resulting from computer experiments. R...

Nyström method has been successfully used to improve the computational efficiency of kernel ridge regression (KRR). Recently, theoretical analysis of Nyström KRR, including generalization bound and convergence rate, has been established based on reproducing kernel Hilbert space (RKHS) associated with the symmetric positive semi-definite kernel. However, in real world applications, RKHS is not a...

Model selection is one of the most important tasks in the identification of blackbox systems. In this paper, we give a novel model selection method from the viewpoint of functional analysis. We formulate the system identification problem as a function approximation problem in a reproducing kernel Hilbert space (RKHS), where the approximation error is measured by the RKHS norm. Within this frame...

Trace norm regularization is a popular method of multitask learning. We give excess risk bounds with explicit dependence on the number of tasks, the number of examples per task and properties of the data distribution. The bounds are independent of the dimension of the input space, which may be infinite as in the case of reproducing kernel Hilbert spaces. A byproduct of the proof are bounds on t...

This paper presents a technique to robustly compare two distributions represented by samples, without explicitly estimating the density. The method is based on mapping the distributions into a reproducing kernel Hilbert space, where eigenvalue decomposition is performed. Retention of only the top M eigenvectors minimizes the effect of noise on density comparison. A sample application of the tec...