Frequently one wants to extend the use of a classification method that in principle requires records with True/False values, so that records with rational numbers can be processed. In such cases, the rational numbers must first be replaced by True/False values before the method may be applied. In other cases, a classification method in principle can process records with rational numbers directl...

Bourgain’s discretization theorem asserts that there exists a universal constant C ∈ (0,∞) with the following property. Let X,Y be Banach spaces with dimX = n. Fix D ∈ (1,∞) and set δ = e−nCn . Assume that N is a δ-net in the unit ball of X and that N admits a bi-Lipschitz embedding into Y with distortion at most D. Then the entire space X admits a bi-Lipschitz embedding into Y with distortion ...

We analytically study the relationship between the Poincaré map and its one step discretization. Error estimates are established depending basically on the right hand side function of the investigated ODE and the given numerical scheme. Our basic tool is a parametric version of a Newton–Kantorovich type methods. As an application, in a neighborhood of a non-degenerate periodic solution a new ty...

We present a generic library which provides means to specify partial differential equations using different discretization schemes, dimensions, and topologies. Due to the common interfaces for simulation domains as well as numerical algebra we have an overall high inter-operability.

For finite difference matrices that are monotone, a discretization error estimate in maximum norm follows from the truncation errors of the discretization. It enables also discretization error estimates for derivatives of the solution. These results are extended to convergent operator splittings of the difference matrix where the major, preconditioning part is monotone but the whole operator is...

Received (Day Month Year) Revised (Day Month Year) We study the discretization of inverse problems defined by a Carleman operator. In particular we develop a discretization strategy for this class of inverse problems and we give a convergence analysis. Learning from examples as well as the discretization of integral equations can be analysed in our setting.

This paper presents an unsupervised discretization method that performs density estimation for univariate data. The subintervals that the discretization produces can be used as the bins of a histogram. Histograms are a very simple and broadly understood means for displaying data, and our method automatically adapts bin widths to the data. It uses the log-likelihood as the scoring function to se...

Discretization is used in click-based graphical passwords so that approximately correct entries can be accepted by the system. We show that the existing discretization scheme of Birget et al.(2006) allows for false accepts and false rejects because the tolerance region is not guaranteed to be centered on the original click-point, causing usability and security concerns. Using empirical data fro...

A new discretization concept for optimal control problems with control constraints is introduced which utilizes for the discretization of the control variable the relation between adjoint state and control. Its key feature is not to discretize the space of admissible controls but to implicitly utilize the first order optimality conditions and the discretization of the state and adjoint equation...

The performance of many machine learning algorithms can be substantially improved with a proper discretization scheme. In this paper we describe a theoretically rigorous approach to discretization of continuous attribute values, based on a Bayesian clustering framework. The method produces a probabilistic scoring metric for diierent discretizations, and it can be combined with various types of ...