Related papers: A robust Khintchine inequality, and algorithms for…

We give new quantum algorithms for evaluating composed functions whose inputs may be shared between bottom-level gates. Let $f$ be an $m$-bit Boolean function and consider an $n$-bit function $F$ obtained by applying $f$ to conjunctions of…

For any $T \geq 1$, there are constants $R=R(T) \geq 1$ and $\zeta=\zeta(T)>0$ and a randomized algorithm that takes as input an integer $n$ and two strings $x,y$ of length at most $n$, and runs in time $O(n^{1+\frac{1}{T}})$ and outputs an…

Let $w_{\alpha}(t)=t^{\alpha}\,e^{-t}$, $\alpha>-1$, be the Laguerre weight function, and $|\cdot|_{w_\alpha}$ denote the associated $L_2$-norm, i.e., $$ | f|_{w_\alpha}:=\Big(\int_{0}^{\infty}w_{\alpha}(t)| f(t)|^2\,dt\Big)^{1/2}. $$…

We give a deterministic algorithm for approximately counting satisfying assignments of a degree-$d$ polynomial threshold function (PTF). Given a degree-$d$ input polynomial $p(x_1,\dots,x_n)$ over $R^n$ and a parameter $\epsilon> 0$, our…

We give an adaptive algorithm which tests whether an unknown Boolean function $f\colon \{0, 1\}^n \to\{0, 1\}$ is unate, i.e. every variable of $f$ is either non-decreasing or non-increasing, or $\epsilon$-far from unate with one-sided…

Grothendieck constants $K_G(d)$ bound the advantage of $d$-dimensional strategies over $1$-dimensional ones in a specific optimisation task. They have applications ranging from approximation algorithms to quantum nonlocality. However, apart…

For a matrix $W \in \mathbb{Z}^{m \times n}$, $m \leq n$, and a convex function $g: \mathbb{R}^m \rightarrow \mathbb{R}$, we are interested in minimizing $f(x) = g(Wx)$ over the set $\{0,1\}^n$. We will study separable convex functions and…

The min-cost matching problem suffers from being very sensitive to small changes of the input. Even in a simple setting, e.g., when the costs come from the metric on the line, adding two nodes to the input might change the optimal solution…

The Gromov-Wasserstein (GW) framework adapts ideas from optimal transport to allow for the comparison of probability distributions defined on different metric spaces. Scalable computation of GW distances and associated matchings on graphs…

Let $\Omega \subset \mathbb{R}^n$ be a convex. If $u: \Omega \rightarrow \mathbb{R}$ has mean 0, then we have the classical Poincar\'{e} inequality $$ \|u \|_{L^p} \leq c_p \mbox{diam}(\Omega) \| \nabla u \|_{L^p}$$ with sharp constants…

Given a multi-variant polynomial inequality with a parameter, how to find the best possible value of this parameter that satisfies the inequality? For instance, find the greatest number $k$ that satisfies $ a^3+b^3+c^3+…

Let \phi(G) be the minimum conductance of an undirected graph G, and let 0=\lambda_1 <= \lambda_2 <=... <= \lambda_n <= 2 be the eigenvalues of the normalized Laplacian matrix of G. We prove that for any graph G and any k >= 2, \phi(G) =…

Computational fluid dynamics (CFD) is a cornerstone of classical scientific computing, and there is growing interest in whether quantum computers can accelerate such simulations. To date, the existing proposals for fault-tolerant quantum…

Let \(Q \subseteq \mathbb{N}\) be a subset, and let \(\psi\colon \mathbb{N} \to [0, \tfrac{1}{2})\), \(\theta\colon \mathbb{N} \to \mathbb{R}\) be functions. Let \(\{A_q\}\) and \(\{B_q\}\) be sequences of integers such that \(\gcd(A_q,…

The Grothendieck constant $K_{G}$ is a fundamental quantity in functional analysis, with important connections to quantum information, combinatorial optimization, and the geometry of Banach spaces. Despite decades of study, the value of…

We study the fundamental problem of learning the parameters of a high-dimensional Gaussian in the presence of noise -- where an $\varepsilon$-fraction of our samples were chosen by an adversary. We give robust estimators that achieve…

This paper presents an improved forward-backward splitting algorithm with two inertial parameters. It aims to find a point in the real Hilbert space at which the sum of a co-coercive operator and a maximal monotone operator vanishes. Under…

Some variant of the Frank-Wolfe method for convex optimization problems with adaptive selection of the step parameter corresponding to information about the smoothness of the objective function (the Lipschitz constant of the gradient).…

We consider the problem of testing whether an unknown Boolean function $f$ is monotone versus $\epsilon$-far from every monotone function. The two main results of this paper are a new lower bound and a new algorithm for this well-studied…

The approximate degree of a Boolean function $f\colon\{0,1\}^n\to\{0,1\}$ is the minimum degree of a real polynomial $p$ that approximates $f$ pointwise: $|f(x)-p(x)|\leq1/3$ for all $x\in\{0,1\}^n.$ For every $\delta>0,$ we construct CNF…