Related papers: AND Testing and Robust Judgement Aggregation
Let $f$ be a function on a bounded domain $\Omega \subseteq \mathbb{R}^n$ and $\delta$ be a positive function on $\Omega$ such that $B(x,\delta(x))\subseteq \Omega$. Let $\sigma(f)(x)$ be the average of $f$ over the ball $B(x,\delta(x))$.…
A classical result in approximation theory states that for any continuous function \( \varphi: \mathbb{R} \to \mathbb{R} \), the set \( \operatorname{span}\{\varphi \circ g : g \in \operatorname{Aff}(\mathbb{R})\} \) is dense in \(…
The classical zero-one law for first-order logic on random graphs says that for any first-order sentence $\phi$ in the theory of graphs, as n approaches infinity, the probability that the random graph G(n, p) satisfies $\phi$ approaches…
For an infinite iterated function system $\mathbf{f}$ on $[0,1]$ with an attractor $\Lambda(\mathbf{f})$ and for an infinite subset $D\subseteq \mathbb{N}$, consider the set \[ \mathbb E(\mathbf{f},D)= \{ x \in \Lambda(\mathbf{f}):…
We discuss some aspects of approximating functions on high-dimensional data sets with additive functions or ANOVA decompositions, that is, sums of functions depending on fewer variables each. It is seen that under appropriate smoothness…
Let $\Sigma_1,\ldots,\Sigma_k$ be finite alphabets, and let $\mu$ be a distribution over $\Sigma_1 \times \dots \times \Sigma_k$ in which the probability of each atom is at least $\alpha$. We prove that if $\mu$ does not admit Abelian…
It is shown that, under certain assumptions on the growth and value distribution of a meromorphic function $f(z)$, \begin{equation*} m\left(r,\frac{\Delta_cf - ac}{f' - a}\right)=S(r,f'), \end{equation*} where $\Delta_c f=f(z+c)-f(z)$ and…
For $0 \leq \beta < \alpha < 1$ the distribution $\mathcal{H}$ over Boolean functions $h \colon \{-1, 1\}^d \to \{-1, 1\}$ that minimizes the expression \begin{equation*} \rho_{\alpha, \beta} = \frac{\log(1/\Pr_{\substack{h \sim \mathcal{H}…
We prove a local contraction property for holomorphic functions that are nearly constant, relating weighted Bergman spaces $A^p_\alpha(\B_n)$ and $A^q_\beta(\B_n)$. Our approach converts geometric information on weighted superlevel sets…
Selman and Kautz's work on ``knowledge compilation'' established how approximation (strengthening and/or weakening) of a propositional knowledge-base can be used to speed up query processing, at the expense of completeness. In this…
In a recent paper, Basu, Hildebrand, and Molinaro established that the set of continuous minimal functions for the 1-dimensional Gomory-Johnson infinite group relaxation possesses a dense subset of extreme functions. The $n$-dimensional…
Two measures of how near an arbitrary function between groups is to being a homomorphism are considered. These have properties similar to conjugates and commutators. The authors show that there is a rich theory based on these structures,…
This is a conitunation of [1] and [2]. We prove that if function $f$ belongs to the class $\Lambda_{\omega} \overset{\text{def}}{=} \{f: \omega_{f}(\delta)\leq \text{const} \omega(\delta)\} $ for an arbitrary modulus of continuity $\omega$,…
Let $\mathbb{F}_p$ be a finite field of prime order $p$ and let $A \subset \mathbb{F}_p$ be a subset. In the dense regime when $|A| \geq \alpha p$ for some $\alpha \in (0,1)$, we determine the optimal constant $f(\alpha)$ in the inequality…
We establish two new direct product theorems for the randomized query complexity of Boolean functions. The first shows that computing $n$ copies of a function $f$, even with a small success probability of $\gamma^n$, requires $\Theta(n)$…
In the field of Judgment Aggrgation, a domain, that is a subset of a Cartesian power of $\{0,1\}$, is considered to reflect abstract rationality restrictions on vectors of two-valued judgments on a number of issues. We are interested in the…
Given a strictly convex multiobjective optimization problem with objective functions $f_1,\dots,f_N$, let us denote by $x_0$ its solution, obtained as minimum point of the linear scalarized problem, where the objective function is the…
Level-1 Consensus is a property of a preference-profile. Intuitively, it means that there exists a preference relation which induces an ordering of all other preferences such that frequent preferences are those that are more similar to it.…
We prove that if $f(n)$ is a Steinhaus or Rademacher random multiplicative function, there almost surely exist arbitrarily large values of $x$ for which $|\sum_{n \leq x} f(n)| \geq \sqrt{x} (\log\log x)^{1/4+o(1)}$. This is the first such…
As the default optimizer for training large language models, AdamW has achieved remarkable success in deep learning. However, its convergence behavior is not theoretically well-understood. This paper establishes the convergence rate…