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We show that if a real x ∈ 2 is strongly Hausdorff Hh-random, where h is a dimension function corresponding to a convex order, then it is also random for a continuous probability measure μ such that the μ-measure of the basic open cylinders shrinks according to h. The proof uses a new method to construct measures, based on effective (partial) continuous transformations and a basis theorem for Π...

The aim of this survey is to collect and explain some geometric results whose proof uses graph or hypergraph theory. No attempt has been made to give a complete list of such results. We rather focus on typical and recent examples showing the power and limitations of the method. The topics covered include forbidden configurations, geometric constructions, saturated hypergraphs in geometry, indep...

Linial et al. raised the question of how difficult the computation of the Vapnik-Červonenkis dimension of a concept class over a finite universe is. Papadimitriou and Yannakakis obtained a first answer using matrix representations of concept classes. However, this approach does not capture classes having exponential size, like monomials, which are encountered in learning theory. We choose a mor...

Recently, the Dimension Problem for effective Hausdorff dimension was solved by J. Miller in [Mil], where the author constructs a Turing degree of non-integral Hausdorff dimension. In this article we settle the Dimension Problem for effective packing dimension by constructing a real of strictly positive effective packing dimension that does not compute a real of effective packing dimension one ...

In multiple-instance learning the learner receives bags, i.e., sets of instances. A bag is labeled positive if it contains a positive example of the target. An Ω(d log r) lower bound is given for the VC-dimension of bags of size r for d-dimensional halfspaces and it is shown that the same lower bound holds for halfspaces over any large point set in general position. This lower bound improves an...

A lower bound of the Hausdorff dimension of certain self-affine sets is given. Moreover, this and other known bounds such as the box dimension are expressed in terms of solutions of simple equations involving the singular values of the affinities. Keyword Codes: G.2.1;G.3

The probably approximately correct (PAC) model of learning and its extension to real-valued function classes sets a rigorous framework based upon which the complexity of learning a target from a function class using a finite sample can be computed. There is one main restriction, however, that the function class have a finite VC-dimension or scale-sensitive pseudo-dimension. In this paper we pre...