The purpose of this paper is to present a largely self-contained proof of the singular value decomposition (SVD), and to explore its application to the low rank approximation problem. We begin by proving background concepts used throughout the paper. We then develop the SVD by way of the polar decomposition. Finally, we show that the SVD can be used to achieve the best low rank approximation of...

For binary classification we establish learning rates up to the order of n−1 for support vector machines (SVMs) with hinge loss and Gaussian RBF kernels. These rates are in terms of two assumptions on the considered distributions: Tsybakov’s noise assumption to establish a small estimation error, and a new geometric noise condition which is used to bound the approximation error. Unlike previous...

About twenty years ago the measure of smoothness ω φ (f, t) was introduced and related to the rate of polynomial approximation. In this article we survey developments about this and related concepts since that time. MSC: 41A10, 41A17, 41A25, 41A27, 41A30, 41A36, 41A40, 41A50, 41A63, 26A15, 26B35, 26B05, 42C05, 26A51, 26A33, 46E35 keywords: Moduli of smoothness, K-functionals, realization functi...

About twenty years ago the measure of smoothness ω φ (f, t) was introduced and related to the rate of polynomial approximation. In this article we survey developments about this and related concepts since that time. MSC: 41A10, 41A17, 41A25, 41A27, 41A30, 41A36, 41A40, 41A50, 41A63, 26A15, 26B35, 26B05, 42C05, 26A51, 26A33, 46E35 keywords: Moduli of smoothness, K-functionals, realization functi...

In this paper, we will first define basic concepts in computational complexity. Then, we will introduce the PCP Theorem and its equivalent formulation from the hardness of approximation view. Finally, we will prove a weaker version of the PCP theorem.

The theoretical concepts for the renormalization of self-consistent Dyson resummations, deviced in the first paper of this series, are applied to first example cases for the φ-theory. Besides the tadpole (Hartree) approximation as a novel part the numerical solutions are presented which includes the sunset self-energy diagram into the self-consistent scheme based on the Φ-derivable approximatio...

For a general infinite system of convex inequalities in a Banach space, we study the basic constraint qualification and its relationship with other fundamental concepts, including various versions of conditions of Slater type, the Mangasarian–Fromovitz constraint qualification, as well as the Pshenichnyi–Levin–Valadier property introduced by Li, Nahak, and Singer. Applications are given in the ...

The mountain car problem is a well-known task, often used for testing reinforcement learning algorithms. It is a problem with real valued state variables, which means that some kind of function approximation is required. In this paper, three reinforcement learning architectures are compared on the mountain car problem. Comparison results are presented, indicating the potentials of the actor-onl...

A rough set is a formal approximation of a crisp set which gives lower and upper approximation of original set to deal with uncertainties. The concept of neutrosophic set is a mathematical tool for handling imprecise, indeterministic and inconsistent data. In this paper, we introduce the concepts of Rough Fuzzy Neutrosophic Sets and Fuzzy Neutrosophic Rough Sets and investigate some of their pr...

Uninorms are important generalizations of triangular norms and conorms, with a neutral element lying anywhere in the unit interval, and left (right) semi-uninorms are non-commutative and non-associative extensions of uninorms. In this paper, we firstly introduce the concepts of left and right semi-uninorms on a complete lattice and illustrate these notions by means of some examples. Then, we la...