Related papers: Renyi Dimension and Gaussian Filtering

We quantify the minimax rate for a nonparametric regression model over a star-shaped function class $\mathcal{F}$ with bounded diameter. We obtain a minimax rate of ${\varepsilon^{\ast}}^2\wedge\mathrm{diam}(\mathcal{F})^2$ where…

After calculating the Dushnik-Miller dimension of Minkowski spaces to be countable infinity, we define a novel notion of dimension for ordered spaces recovering the correct manifold dimension and obtain a corresponding obstruction for the…

We identify shortcomings in two popular measures of localization of functions: the $L^p-L^q$ participation ratio and the mass concentration comparison. We then introduce a novel localization measure for functions on bounded subsets of…

Let $(\mathcal{X},\rho)$ be a metric space and $\lambda$ be a Borel measure on this space defined on the $\sigma$-algebra generated by open subsets of $\mathcal{X}$; this measure $\lambda$ defines volumes of Borel subsets of $\mathcal{X}$.…

Let $m,n\in\mathbb{N}$ and $p\in(0,\infty)$. For a finite dimensional quasi-normed space $X=(\mathbb{R}^m, \|\cdot\|_X)$, let $$B_p^n(X) = \Big\{ (x_1,\ldots,x_n)\in\big(\mathbb{R}^{m}\big)^n: \ \sum_{i=1}^n \|x_i\|_X^p \leq 1\Big\}.$$ We…

We consider the problem of minimizing a continuous function f over a compact set K. We analyze a hierarchy of upper bounds proposed by Lasserre in [SIAM J. Optim. 21(3) (2011), pp. 864--885], obtained by searching for an optimal probability…

We evaluate partition functions of matrix models which are given by topologically twisted and dimensionally reduced actions of d=4 N=1 super Yang-Mills theories with classical (semi-)simple gauge groups, SO(2N), SO(2N+1) and USp(2N). The…

We discuss new proofs, and new forms, of a reverse logarithmic Sobolev inequality, with respect to the standard Gaussian measure, for low complexity functions, measured in terms of Gaussian-width. In particular, we provide a dimension-free…

The problem of estimating, from a random sample of points, the dimension of a compact subset $S$ of the Euclidean space is considered. The emphasis is put on consistency results in the statistical sense. That is, statements of convergence…

In this article, we study the smallest distances between the zeros of Gaussian analytic functions over compact Riemann surfaces. Our main result is that, after appropriate rescaling, the point process of the smallest distances converge to a…

We give an upper bound for the error term in the Hardy-Ramanujan-Rademacher formula for the partition function. The main input is a new hybrid subconvexity bound for the central value $L(\tfrac 12,f\times (\tfrac{q}{\cdot}))$ in the $q$ and…

A randomized scheme that succeeds with probability $1-\delta$ (for any $\delta>0$) has been devised to construct (1) an equidistributed $\epsilon$-cover of a compact Riemannian symmetric space $\mathbb M$ of dimension $d_{\mathbb M}$ and…

We prove a ''dimension expansion'' version of the Elekes-R\'onyai theorem for trivariate real analytic functions: If $f$ is a trivariate real analytic function, then $f$ is either locally of the form $g(h(x)+k(y)+l(z))$, or the following is…

We prove that for any semi-Dirichlet form $(\epsilon, D(\epsilon))$ on a measurable Lusin space $E$ there exists a Lusin topology with the given $\sigma$-algebra as the Borel $\sigma$-algebra so that $(\epsilon, D(\epsilon))$ becomes…

Given a subset K of the unit Euclidean sphere, we estimate the minimal number m = m(K) of hyperplanes that generate a uniform tessellation of K, in the sense that the fraction of the hyperplanes separating any pair x, y in K is nearly…

In the quantum theory, using the notion of partial supersymmetry, in which some, but not all, operators have superpartners we derive the Euler theorem in partition theory. The paraferminic partition function gives another identity in…

We provide upper bounds of the expected Wasserstein distance between a probability measure and its empirical version, generalizing recent results for finite dimensional Euclidean spaces and bounded functional spaces. Such a generalization…

Building on work of Hardy and Ramanujan, Rademacher proved a well-known formula for the values of the ordinary partition function $p(n)$. More recently, Bruinier and Ono obtained an algebraic formula for these values. Here we study the…

Density of Lipschitz functions in Newtonian spaces based on quasi-Banach function lattices is discussed. Newtonian spaces are first-order Sobolev-type spaces on abstract metric measure spaces defined via (weak) upper gradients. Our main…

A celebrated result in convex geometry is Gr\"unbaum's inequality, which quantifies how much volume of a convex body can be cut off by a hyperplane passing through its barycenter. In this work, we establish a series of sharp Gr\"unbaum-type…