The goal of these notes is to introduce the reader to the Heisenberg group with its CarnotCarathéodory metric and to Pansu’s differentiation theorem. As they are very similar, we will first study Rademacher’s theorem about Lipschitz maps and then see how the same technique can be applied in the more complex setting of the Heisenberg group.

Let p(n) denote the classical partition function, defined as the number of ways of writing n as a sum of positive integers, where the order of the summands is not taken into account. Hardy and Ramanujan [10] developed the circle method in order to obtain precise estimates of the asymptotic behavior of p(n) as n!1. Their results were later refined by Rademacher [13], who found an exact expressio...

We introduce the notion of cotype of a metric space, and prove that for Banach spaces it coincides with the classical notion of Rademacher cotype. This yields a concrete version of Ribe's theorem, settling a long standing open problem in the non-linear theory of Banach spaces. We apply our results to several problems in metric geometry. Namely, we use metric cotype in the study of uniform and c...

We investigate a new methodology, worked out by Ledoux and Mas-sart, to prove concentration-of-measure inequalities. The method is based on certain modified logarithmic Sobolev inequalities. We provide some very simple and general ready-to-use inequalities. One of these inequalities may be considered as an exponential version of the Efron-Stein inequality. The main purpose of this paper is to p...

with probability at least 1−δ. The quantity Rn(F (Z n)) appearing on the right-hand side of the above bound is the Rademacher average of the random set F (Z ) = {( f (Z1), . . . , f (Zn)) : f ∈F} , often referred to as the projection of F onto the sample Z n . From this we see that a sufficient condition for the ERM algorithm to produce near-optimal hypotheses with high probability is that the ...

We study the elliptic genera of hyperKähler manifolds using the representation theory of N = 4 superconformal algebra. We consider the decomposition of the elliptic genera in terms of N = 4 irreducible characters, and derive the rate of increase of the multiplicities of half-BPS representations making use of Rademacher expansion. Exponential increase of the multiplicity suggests that we can ass...

Sequential algorithms of active learning based on the estimation of the level sets of the empirical risk are discussed in the paper. Localized Rademacher complexities are used in the algorithms to estimate the sample sizes needed to achieve the required accuracy of learning in an adaptive way. Probabilistic bounds on the number of active examples have been proved and several applications to bin...

A.2. Contraction lemma The following is a version of Talagrand’s contraction lemma (Ledoux & Talagrand, 2011). Since our definition of Rademacher complexity does not use absolute values, we give an explicit proof below. Lemma 8. Let H be a hypothesis set of functions mapping X to R and Ψ1, . . . ,Ψm, μ-Lipschitz functions for some μ > 0. Then, for any sample S of m points x1, . . . , xm ∈ X , t...