Related papers: On the Complexity of Binary Samples

An $s$-subset of codewords of a binary code $X$ is said to be an {\em $(s,\ell)$-bad} in $X$ if the code $X$ contains a subset of other $\ell$ codewords such that the conjunction of the $\ell$ codewords is covered by the disjunctive sum of…

We use deep sparsely connected neural networks to measure the complexity of a function class in $L^2(\mathbb R^d)$ by restricting connectivity and memory requirement for storing the neural networks. We also introduce representation system -…

We formalize the observation that the same summability methods converge in a Banach space $X$ and its dual $X^*$. At the same time we determine conditions under which these methods converge in the weak and weak*-topologies on $X$ and $X^*$…

Here is one of the results of this paper (with the convention ${{1}\over {0}}=+\infty$): Let $X$ be a real Hilbert space and let $J:X\to {\bf R}$ be a $C^1$ functional, with compact derivative, such that $$\alpha^*:=\max\left…

Specimens are collected from $N$ different sources. Each specimen has probability $p$ of being contaminated, independently of the other specimens. We assume group testing is applicable, namely one can take small portions from several…

We investigate subsets with small sumset in arbitrary abelian groups. For an abelian group $G$ and an $n$-element subset $Y \subseteq G$ we show that if $m \ll s^2/(\log n)^2$, then the number of subsets $A \subseteq Y$ with $|A| = s$ and…

In this paper, we extend the correspondence between Bayesian estimation and optimal smoothing in a Reproducing Kernel Hilbert Space (RKHS) adding a convexe constraints on the solution. Through a sequence of approximating Hilbertian spaces…

Sup-norm curve estimation is a fundamental statistical problem and, in principle, a premise for the construction of confidence bands for infinite-dimensional parameters. In a Bayesian framework, the issue of whether the…

Let $A=\{a_0,a_1,\ldots,a_{k-1}\}$ be a set of $k$ integers. For any integer $h\ge 1$ and any ordered $k$-tuple of positive integers $\mathbf{r}=(r_0,r_1,\ldots,r_{k-1})$, we define a general $h$-fold sumset, denoted by $h^{(\mathbf{r})}A$,…

Let $N$ be a finite set of cardinality $n$, and $a\in N$. A submodular function $f$ on $N$ with $f(a)=1$ is defined to be $a$-reduced if, for any decomposition $f=g+h$ into submodular functions where $h$ does not depend on $a$, it follows…

Let $D$ be a dictionary in a Hilbert space $H$, that is, a set of unit elements whose linear combinations are dense in $H$. We consider the least $m$-term deviation $\sigma_m(x)$ of an element $x\in H$: this is the distance of $x$ from the…

For a finite set $A\subseteq \mathbb{Z}$, the $h$-fold sumset is $hA :=\{x_1+\dots+x_h:x_i\in A\}$. We interpret the beginning of the sequence of sumset sizes $(|hA|)_{h=1}^\infty$ in terms of the successive $L^1$-minima of a lattice…

Sample size criteria are often expressed in terms of the concentration of the posterior density, as controlled by some sort of error bound. Since this is done pre-experimentally, one can regard the posterior density as a function of the…

We establish a regular sampling theory in the range of the analysis operator of a continuous frame having a unitary structure. The unitary structure is related with a unitary representation of a locally compact abelian group on a separable…

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 define the notion of {\em rational presentation of a complete metric space} in order to study metric spaces from the algorithmic complexity point of view. In this setting, we study some presentations of the space $\czu$ of uniformly…

Given a Hilbert space $\mathcal H$ and a finite measure space $\Omega$, the approximation of a vector-valued function $f: \Omega \to \mathcal H$ by a $k$-dimensional subspace $\mathcal U \subset \mathcal H$ plays an important role in…

In this work, we analyze the learnability of reproducing kernel Hilbert spaces (RKHS) under the $L^\infty$ norm, which is critical for understanding the performance of kernel methods and random feature models in safety- and…

In the framework of a strictly local regular Dirichlet space ${\bf X}$ we introduce the subspaces $PW_{\omega},\>\>\omega>0,$ of Paley-Wiener functions of bandwidth $\omega$. It is shown that every function in $PW_{\omega},\>\>\omega>0,$ is…

We improve the universality theorem of the Riemann zeta-function in short intervals by establishing universality for significantly shorter intervals $[T,T+H]$. Assuming the Riemann Hypothesis, we prove that universality in such short…