Related papers: On the Complexity of Binary Samples

We introduce and study a new topological notion of the size for subsets of the real line, called \emph{super-density}. A set $A\subset\mathbb{R}$ is super-dense if for every non-empty open interval $I$ and every nowhere constant continuous…

A $k$-configuration is a collection of $k$ distinct integers $x_1,\ldots,x_k$ together with their pairwise arithmetic means $\frac{x_i+x_j}{2}$ for $1 \leq i < j \leq k$. Building on recent work of Filmus, Hatami, Hosseini and Kelman on…

We show that there are sampling projections on arbitrary $n$-dimensional subspaces of $B(D)$ with at most $2n$ samples and norm of order $\sqrt{n}$, where $B(D)$ is the space of complex-valued bounded functions on a set $D$. This gives a…

This paper resolves two open problems from a recent paper, arXiv:2403.16981, concerning the sample complexity of distributed simple binary hypothesis testing under information constraints. The first open problem asks whether interaction…

We study the density function of measurable subsets of the Cantor space. Among other things, we identify a universal set $\mathcal{U}$ for $\Sigma^{1}_{1}$ subsets of $( 0 ; 1 )$ in terms of the density function; specifically $\mathcal{U}$…

We initiate the rigorous study of classification in semimetric spaces, which are point sets with a distance function that is non-negative and symmetric, but need not satisfy the triangle inequality. For metric spaces, the doubling dimension…

Let $X=C[0,1]$, and $Y$ be an arbitrary Banach space. Consider a collection of open segments $\{V_i \}\subset X$. Suppose the map $f: \cup_i V_i \to Y$ has $q$ bounded Fr\'echet derivatives ($q=0,1,...,\infty$), and $f$ and all its…

A subset $A$ of a commutative semigroup $X$ is called a $B_h$ set in $X$ if the only solutions to $a_1+\dots+a_h = b_1 + \cdots +b_h$ (with $a_i,b_i \in A$) are the trivial solutions $\{a_1,\dots,a_h\} = \{b_1,\dots,b_h\}$ (as multisets).…

It is an interesting, maybe surprising, fact that different dense subspaces of even "nice" topological spaces can have different densities. So, our aim here is to investigate the set of densities of all dense subspaces of a topological…

We propose an incremental strategy for learning hash functions with kernels for large-scale image search. Our method is based on a two-stage classification framework that treats binary codes as intermediate variables between the feature…

We propose a novel test procedure for comparing mean functions across two groups within the reproducing kernel Hilbert space (RKHS) framework. Our proposed method is adept at handling sparsely and irregularly sampled functional data when…

Let $\mathscr{H}^\infty$ be the set of all Dirichlet series $f=\sum\limits_{n=1}^\infty \frac{a_n}{n^s}$ (where $a_n\in \mathbb{C}$ for each $n$) that converge at each $s\in {\mathbb{C}}_+$, such that $\|f\|_{\infty}:=\sup_{s\in…

We show that if $X$ is a Banach space with a Schauder basis, $\Omega\subset X$ is a pseudoconvex open subset, and $u:\,\Omega\to(-\infty,\infty)$ is a locally bounded function, then there are a Banach space $Z$ and a holomorphic function…

The problem of establishing out-of-sample bounds for the values of an unkonwn ground-truth function is considered. Kernels and their associated Hilbert spaces are the main formalism employed herein along with an observational model where…

We consider the density $X_t(x)$ of the critical $(\alpha,\beta)$-superprocess in $R^d$ with $\alpha\in (0,2)$ and $\beta<\frac \alpha d$. A recent result from PDE implies a dichotomy for the density: for fixed $x$, $X_t(x)>0$ a.s. on…

Some properties of $m$-density points and density-degree functions are studied. Moreover the following main results are provided: \vskip2mm \begin{itemize} \item {\it Let $\lambda$ be a continuous differential form of degree $h$ in…

Given a subset $\Lambda$ of $\mathbb Z_+:=\{0,1,2,\dots\}$, let $H^\infty(\Lambda)$ denote the space of bounded analytic functions $f$ on the unit disk whose coefficients $\widehat f(k)$ vanish for $k\notin\Lambda$. Assuming that either…

Let $S = \{p_1,\dots,p_r\}$ be a finite set of distinct primes, let $\Psi_S(X)$ be the number of $S$-smooth integers not exceeding $X$, and let $F_S(X)$ be the maximum size of a subset of $M(S) \cap [1,X]$ containing no set $\{n,p_1…

We study the behavior of the posterior distribution in high-dimensional Bayesian Gaussian linear regression models having $p\gg n$, with $p$ the number of predictors and $n$ the sample size. Our focus is on obtaining quantitative finite…

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…