We introduce elliptic analogues to the Bernoulli ( resp. Euler) numbers and functions. The first aim of this paper is to state and prove that our elliptic Bernoulli and Euler functions satisfied Raabe’s formulas (cf. Theorems 3.1.1, 3.2.1). We define two kinds of elliptic Dedekind-Rademacher sums, in terms of values of our elliptic Bernoulli (resp. Euler) functions. The second aim of this paper...

This paper revisits a problem of the evaluation of computational grammatical inference (GI) systems and discusses what role complexity measures can play for the assessment of GI. We provide a motivation for using the Rademacher complexity and give an example showing how this complexity measure can be used in practice.

Rademacher averages have been proposed as data-dependent estimates of error in pattern classification and regression problems. While they exploit information about the probability distribution that generates the data, they give pessimistic estimates in many cases. For instance, when zero error is possible, these estimates have a suboptimal convergence rate. We investigate the use of local Radem...

Boltzmann machine, as a fundamental construction block of deep belief network and deep Boltzmann machines, is widely used in deep learning community and great success has been achieved. However, theoretical understanding of many aspects of it is still far from clear. In this paper, we studied the Rademacher complexity of both the asymptotic restricted Boltzmann machine and the practical impleme...

We study decoupling in quasi-Banach spaces. We show that decoupling is permissible in some quasi-Banach spaces (e.g., Lp and Lp/Hp when 0 < p < 1) but fails in other spaces such as the Schatten ideal Sp when 0 < p < 1. We also relate our ideas to a possible extension of the Grothendieck inequality.

We develop a new family of algorithms for the online learning setting with regret against any data sequence bounded by the empirical Rademacher complexity of that sequence. To develop a general theory of when this type of adaptive regret bound is achievable we establish a connection to the theory of decoupling inequalities for martingales in Banach spaces. When the hypothesis class is a set of ...

We investigate the fat-shattering dimension and the localized Rademacher averages of kernel machines and their connection to the eigenvalues associated with the kernel.