December 22, 1999 Abstract. Suppose A is a hyperfinite von Neumann algebra with a normal faithful normalized trace τ . We prove that if E is a homogeneous Hilbertian subspace of Lp(τ) (1 ≤ p < ∞) such that the norms induced on E by Lp(τ) and L2(τ) are equivalent, then E is completely isomorphic to the subspace of Lp([0, 1]) spanned by Rademacher functions. Consequently, any homogeneous Hilberti...

A collection of Hardy-Ramanujan-Rademacher type formulas for restricted partition and overpartition functions is presented, framed by several biographical anecdotes.

For non-reversible Finsler metrics of positive flag curvature on spheres and projective spaces we present results about the number and the length of closed geodesics and about their stability properties.

We prove a minimum degree version of the Kruskal--Katona theorem for triple systems: given $d\ge 1/4$ and system $\mathcal{F}$ on $n$ vertices with $\delta(\mathcal{F})\ge d\binom n2$, we obtain asymptotically tight lower bounds size its shadow. Equivalently, $t\ge n/2-1$, determine graph vertices, in which every vertex is contained at least $\binom t2$ triangles. This can be viewed as variant ...

This paper studies the problem of learning weighted automata from a finite sample of strings with real-valued labels. We consider several hypothesis classes of weighted automata defined in terms of three different measures: the norm of an automaton’s weights, the norm of the function computed by an automaton, and the norm of the corresponding Hankel matrix. We present new data-dependent general...

j=1 hk,j |8(k, j) 2 [p]⇥[Nk], hk,j 2Hk , and the union of all such families GF,n = S |N|=n GF,N. Fix ⇢ > 0. For a fixed N, the Rademacher complexity of GF,N can be bounded as follows for any m 1: Rm(GF,N)  1 n Pp k=1 Nk Rm(Hk). Thus, the following standard margin-based Rademacher complexity bound holds (Koltchinskii & Panchenko, 2002). For any > 0, with probability at least 1 , for all g 2 GF,...

Great successes of deep neural networks have been witnessed in various real applications. Many algorithmic and implementation techniques have been developed; however, theoretical understanding of many aspects of deep neural networks is far from clear. A particular interesting issue is the usefulness of dropout, which was motivated from the intuition of preventing complex co-adaptation of featur...

i=1 ǫiAi (1) of deterministic Hermitian matrices A1, . . . , An multiplied by random coefficients. Recall that a Rademacher sequence is a sequence {ǫi}i=1 of i.i.d. random variables with ǫ1 uniform over {−1,+1}. A standard Gaussian sequence is a sequence i.i.d. standard Gaussian random variables. Our main goal is to prove the following result. Theorem 1 (proven in Section 3) Given positive inte...

Abstract We investigate the use of certain data-dependent estimates of the complexity of a function class, called Rademacher and Gaussian complexities. In a decision theoretic setting, we prove general risk bounds in terms of these complexities. We consider function classes that can be expressed as combinations of functions from basis classes and show how the Rademacher and Gaussian complexitie...