This work studies the convergence and finite sample approximations of entropic regularized Wasserstein distances in Hilbert space setting. Our first main result is that for Gaussian measures on an infinite-dimensional space, 2-Sinkhorn divergence strictly weaker than exact 2-Wasserstein distance. Specifically, a sequence centered converges if corresponding covariance operators converge Hilbert–...

and Applied Analysis 3 2. Preliminaries Let w α,β x 1 − x α 1 x β be a weight function in the usual sense for α, β > −1. The set of Jacobi polynomials {P α,β k x } k 0 forms a complete L 2 w α,β −1, 1 -orthogonal system, and ∥ ∥ ∥P α,β k ∥ ∥ ∥ 2 w α,β h α,β k 2 β 1Γ k α 1 Γ ( k β 1 ) ( 2k α β 1 ) Γ k 1 Γ ( k α β 1 ) . 2.1 Here, L2 w α,β −1, 1 is a weighted space defined by L2 w α,β −1, 1 {v : v...

Given the firstand kth-order rate functions, where k ≥ 2, we define gk(t1, · · · , tk;Ni) = λk(t1, · · · , tk;Ni) λ(t1;Ni) · · ·λ(tk;Ni) . Throughout this section, let c, c1 and c2 denote three positive real numbers where c1 < c2. Let β0 denote the true but unknown value of β. Let ρ (1) i (β) denote the vector of partial derivatives of ρi(β) with respect to β. Let || · || be the Euclidean norm....

In this paper, we prove that for l = 1 or 2 the rate of best l-monotone polynomial approximation in the Lp norm (1 ≤ p ≤ ∞) weighted by the Jacobi weight wα ,β(x) ∶= (1 + x)α(1 − x)β with α, β > −1/p if p <∞, or α, β ≥ 0 if p =∞, is bounded by an appropriate (l + 1)-st modulus of smoothness with the same weight, and that this rate cannot be bounded by the (l+2)-ndmodulus. Related results on con...

Classification algorithms based on different forms of support vector machines (SVMs) for dealing with interval-valued training data are proposed in the paper. L2-norm and L∞-norm SVMs are used for constructing the algorithms. The main idea allowing us to represent the complex optimization problems as a set of simple linear or quadratic programming problems is to approximate the Gaussian kernel ...

We introduce the fractional mixed fractional Brownian sheet and investigate the small ball behavior of its sup-norm statistic by establishing a general result on the small ball probability of the sum of two not necessarily independent joint Gaussian random vectors. Then, we state general conditions and characterize the sufficiency part of the lower classes of some statistics of the above proces...

We show that support vector machines of the 1-norm soft margin type are universally consistent provided that the regularization parameter is chosen in a distinct manner and the kernel belongs to a specific class}the so-called universal kernels}which has recently been considered by the author. In particular it is shown that the 1-norm soft margin classifier with Gaussian RBF kernel on a compact ...

We investigate selfadjoint positivity preserving C0-semigroups that are dominated by the free heat semigroup on Rd. Major examples are semigroups generated by Dirichlet Laplacians on open subsets or by Schrödinger operators with absorption potentials. We show explicit global Gaussian upper bounds for the kernel that correctly reflect the exponential decay of the semigroup. For eigenfunctions of...