Convexity and logical analysis of data

Abstract

Cataloged from PDF version of article.A Boolean function is called k-convex if for any pair x,y of its true points at Hamming distance at most k, every point "between" x and y is also true. Given a set of true points and a set of false points, the central question of Logical Analysis of Data is the study of those Boolean functions whose values agree with those of the given points. In this paper we examine data sets which admit k-convex Boolean extensions. We provide polynomial algorithms for finding a k-convex extension, if any, and for finding the maximum k for which a k-convex extension exists. We study the problem of uniqueness, and provide a polynomial algorithm for checking whether all k-convex extensions agree in a point outside the given data set. We estimate the number of k-convex Boolean functions, and show that for small k this number is doubly exponential. On the other hand, we also show that for large k the class of k-convex Boolean functions is PAC-learnable. (C) 2000 Elsevier Science B.V. All rights reserved

Prediction of Protein Secondary Structure Using Logical Analysis of Data Algorithm

“…Because of a complexity of the algorithm of Logical Analysis of Data [9] it is hard to present all aspects of this method. All important phases one can see in Fig.…”

Logical analysis of data – the vision of Peter L. Hammer

“…While a decade ago the focus of research was on theoretical developments (see e.g., [21,31,32,40]) and on generic computational implementation [22], in recent years Peter L. Hammer's attention was concentrated on practical applications, particularly to medical problems, which started in 2002-2003 with the publication of the conclusions of a collaborative study with medical researchers at the Cleveland Clinic Foundation on risk assessment among cardiac patients [9,46].…”

Learning Weakly Convex Sets in Metric Spaces

“…More precisely, a subset A of a metric space is weakly convex if for all x, y ∈ A and for all points z in the ground set, z belongs to A whenever x and y are near to each other and the three points satisfy the triangle inequality with equality. This definition has been inspired by the following relaxation of convexity in the Hamming metric space [6]: A Boolean function is k-convex for some positive integer k if for all true points x and y having a Hamming distance of at most k, all points on all shortest paths between x and y are also true. Our definition of weak convexity generalizes this notion to arbitrary metric spaces.…”