Given a set of classifiers and a probability distribution over their domain, one can define a metric by taking the distance between a pair of classifiers to be the probability that they classify a random item differently. We prove bounds on the sample complexity of PAC learning in terms of the doubling dimension of this metric. These bounds imply known bounds on the sample complexity of learnin...

Using Voiculescu’s notion of a matricial microstate we introduce fractal dimensions and entropies for finite sets of selfadjoint operators in a tracial von Neumann algebra. We show that they possess properties similar to their classical predecessors. We relate the new quantities to free entropy and free entropy dimension and show that a modified version of free Hausdorff dimension is an algebra...

Multi-fractal patterns occur widely in nature. In developing new algorithms to determine multi-fractal spectra of experimental data I am lead to the conclusion that generalised dimensions Dq of order q ≤ 0, including the Hausdorff dimension, are effectively irrelevant. The reason is that these dimensions are extraordinarily sensitive to regions of low density in the multi-fractal data. Instead,...

The construction of codes in the projective space for error control in random networks has been the focus of recent research. The Plücker coordinate description of subspaces has been discussed in the context of constant dimension codes, as well as the Schubert cell description of certain code parameters. In this paper we use this classical tool to reformulate some standard constructions of cons...