The covering number for some Mercer kernel Hilbert spaces

AbstractIn the present paper, we investigate the estimates for the covering number of a ball in a Mercer kernel Hilbert space on [0,1]. Let Pl(x) be the Legendre orthogonal polynomial of order l, al>0 be real numbers satisfying ∑l=0+∞lal<+∞. Then, for the Mercer kernel functionK(x,t)=∑l=0+∞alPl(x)Pl(t),x,t∈[0,1],we provide the upper estimates of the covering number for the Mercer kernel Hilbert space reproducing from K(x,t). For some particular al we give the lower estimates. Meanwhile, a kind of l2-norm estimate for the inverse Mercer matrix associated with the Mercer kernel K(x,t) is given

Extensions of Positive Definite Functions: Applications and Their Harmonic Analysis

We study two classes of extension problems, and their interconnections: (i) Extension of positive definite (p.d.) continuous functions defined on subsets in locally compact groups $G$; (ii) In case of Lie groups, representations of the associated Lie algebras $La\left(G\right)$ by unbounded skew-Hermitian operators acting in a reproducing kernel Hilbert space (RKHS) $\mathscr{H}_{F}$. Why extensions? In science, experimentalists frequently gather spectral data in cases when the observed data is limited, for example limited by the precision of instruments; or on account of a variety of other limiting external factors. Given this fact of life, it is both an art and a science to still produce solid conclusions from restricted or limited data. In a general sense, our monograph deals with the mathematics of extending some such given partial data-sets obtained from experiments. More specifically, we are concerned with the problems of extending available partial information, obtained, for example, from sampling. In our case, the limited information is a restriction, and the extension in turn is the full positive definite function (in a dual variable); so an extension if available will be an everywhere defined generating function for the exact probability distribution which reflects the data; if it were fully available. Such extensions of local information (in the form of positive definite functions) will in turn furnish us with spectral information. In this form, the problem becomes an operator extension problem, referring to operators in a suitable reproducing kernel Hilbert spaces (RKHS). In our presentation we have stressed hands-on-examples. Extensions are almost never unique, and so we deal with both the question of existence, and if there are extensions, how they relate back to the initial completion problem.Comment: 235 pages, 42 figures, 7 tables. arXiv admin note: substantial text overlap with arXiv:1401.478

Learning Sets with Separating Kernels

We consider the problem of learning a set from random samples. We show how relevant geometric and topological properties of a set can be studied analytically using concepts from the theory of reproducing kernel Hilbert spaces. A new kind of reproducing kernel, that we call separating kernel, plays a crucial role in our study and is analyzed in detail. We prove a new analytic characterization of the support of a distribution, that naturally leads to a family of provably consistent regularized learning algorithms and we discuss the stability of these methods with respect to random sampling. Numerical experiments show that the approach is competitive, and often better, than other state of the art techniques.Comment: final versio