In this paper, we explore two conjectures about Rademacher sequences. Let (εi) be a sequence, i.e., sequence of independent {−1, 1}-valued symmetric random variables. Set Sn = a1ε1 + … anεn for (a1, …, an) ∈ ℝn. The rst conjecture says that P (|Sn| ≤ ‖a‖) ≥ \({\mathbb{R}^n}\) all ℝn and n \(\mathbb{N}\). second ℕ. Regarding the first conjecture, present several new equivalent formulations. Thes...

We present a novel notion of complexity that interpolates between and generalizes some classic existing complexity notions in learning theory: for estimators like empirical risk minimization (ERM) with arbitrary bounded losses, it is upper bounded in terms of data-independent Rademacher complexity; for generalized Bayesian estimators, it is upper bounded by the data-dependent information comple...

For 0 < a < 1, let/a(x) = 2,1,2~ Ri(x) for 0 s$ x < 1, where {#,}£, is the sequence of Rademacher functions. We give a class of fa so that their graphs have Hausdorff dimension 2 —a. The result is closely related to the corresponding unsolved question for the Weierstrass functions.

A Rademacher-type convergent series formula which generalizes the Hardy-Ramanujan-Rademacher formula for the number of partitions of n and the Zuckerman formula for the Fourier coefficients of θ4(0 | τ)−1 is presented. 1. Background 1.1. Partitions. A partition of an integer n is a representation of n as a sum of positive integers, where the order of the summands (called parts) is considered ir...

We show that a Rademacher expansion can be used to establish an exact bound for the entropy of black holes within a conformal field theory framework. This convergent expansion includes all subleading corrections to the Bekenstein-Hawking term.

We study the problem of learning ensembles in the online setting, when the hypotheses are selected out of a base family that may be a union of possibly very complex sub-families. We prove new theoretical guarantees for the online learning of such ensembles in terms of the sequential Rademacher complexities of these sub-families. We also describe an algorithm that benefits from such guarantees. ...