Discrepancy norm: approximation and variations

Abstract This paper introduces an approach for the minimization of the discrepancy norm. The general idea is to replace the infinity norms appearing in the definition by L p norms which are differentiable and to make use of this approximation for local optimization. We will show that the discrepancy norm can be approximated up to any ε and the robustness of this approximation is shown. Moreover, analytical formulation of the derivative of the discrepancy correlation function is given. In a following step we extend the results to higher dimensional data and derive the related forms for approximations and differentiations

Spectral Analysis of the MIXMAX Random Number Generators

International audienceWe study the lattice structure of random number generators of the MIXMAX family, a class of matrix linear congruential generators that produce a vector of random numbers at each step. These generators were initially proposed and justified as close approximations to certain ergodic dynamical systems having the Kolmogorov K-mixing property, which implies a chaotic (fast-mixing) behavior. But for a K-mixing system, the matrix must have irrational entries, whereas for the MIXMAX it has only integer entries. As a result, the MIXMAX has a lattice structure just like linear congruential and multiple recursive generators. Its matrix entries were also selected in a special way to allow a fast implementation and this has an impact on the lattice structure. We study this lattice structure for vectors of successive and non-successive output values in various dimensions. We show in particular that for coordinates at specific lags not too far apart, in three dimensions, all the nonzero points lie in only two hyperplanes. This is reminiscent of the behavior of lagged-Fibonacci and AWC/SWB generators. And even if we skip the output coordinates involved in this bad structure, other highly structured projections often remain, depending on the choice of parameters. We show that empirical statistical tests can easily detect this structure

Spectral analysis of random number generators

This paper is based on the theory developed by Dr. Evangelos Yfantis, professor of Computer Science at University of Nevada, Las Vegas. In this paper, we describe a method for testing the fairness of pseudorandom number generators using the Discrete Fourier Transform. We will show how the concept of a random process can be used in a representation for random discrete time signals. Using this concept, we have focused on the mathematical representations of the spectral analysis of a fair pseudorandom number generator. From this representation, a reasonable spectral expectation is determined. An algorithm which applies the developed method is described, and a modified shift register random number generator is used to produce sample data

Kullback-Leibler information function and the sequential selection of experiments to discriminate among several linear models

A sequential adaptive experimental design procedure for a related problem is studied. It is assumed that a finite set of potential linear models relating certain controlled variables to an observed variable is postulated, and that exactly one of these models is correct. The problem is to sequentially design most informative experiments so that the correct model equation can be determined with as little experimentation as possible. Discussion includes: structure of the linear models; prerequisite distribution theory; entropy functions and the Kullback-Leibler information function; the sequential decision procedure; and computer simulation results. An example of application is given

On the hardness of the shortest vector problem

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 1998.Includes bibliographical references (p. 77-84).An n-dimensional lattice is the set of all integral linear combinations of n linearly independent vectors in Rm. One of the most studied algorithmic problems on lattices is the shortest vector problem (SVP): given a lattice, find the shortest non-zero vector in it. We prove that the shortest vector problem is NP-hard (for randomized reductions) to approximate within some constant factor greater than 1 in any 1, norm (p >\=1). In particular, we prove the NP-hardness of approximating SVP in the Euclidean norm 12 within any factor less than [square root of]2. The same NP-hardness results hold for deterministic non-uniform reductions. A deterministic uniform reduction is also given under a reasonable number theoretic conjecture concerning the distribution of smooth numbers. In proving the NP-hardness of SVP we develop a number of technical tools that might be of independent interest. In particular, a lattice packing is constructed with the property that the number of unit spheres contained in an n-dimensional ball of radius greater than 1 + [square root of] 2 grows exponentially in n, and a new constructive version of Sauer's lemma (a combinatorial result somehow related to the notion of VC-dimension) is presented, considerably simplifying all previously known constructions.by Daniele Micciancio.Ph.D