Related papers: An Improved Line-Point Low-Degree Test
A perfect matching in an undirected graph $G=(V,E)$ is a set of vertex disjoint edges from $E$ that include all vertices in $V$. The perfect matching problem is to decide if $G$ has such a matching. Recently Rothvo{\ss} proved the striking…
Weighted least squares polynomial approximation uses random samples to determine projections of functions onto spaces of polynomials. It has been shown that, using an optimal distribution of sample locations, the number of samples required…
The Schm\"udgen's Positivstellensatz gives a certificate to verify positivity of a strictly positive polynomial $f$ on a compact, basic, semi-algebraic set $\mathbf{K} \subset \mathbb{R}^n$. A Positivstellensatz of this type is called…
We study the last fall degrees of {\em semi-local} polynomial systems, and the computational complexity of solving such systems for closed-point and rational-point solutions, where the systems are defined over a finite field. A semi-local…
A fault-tolerant distance labeling scheme assigns a label to each vertex and edge of an undirected weighted graph $G$ with $n$ vertices so that, for any edge set $F$ of size $|F| \leq f$, one can approximate the distance between $p$ and $q$…
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…
Certifiable local robustness, which rigorously precludes small-norm adversarial examples, has received significant attention as a means of addressing security concerns in deep learning. However, for some classification problems, local…
We develop a new algorithm for factoring a bivariate polynomial $F\in \mathbb{K}[x,y]$ which takes fully advantage of the geometry of the Newton polygon of $F$. Under a non degeneracy hypothesis, the complexity is…
In this paper, we investigate property testing whether or not a degree d multivariate poly- nomial is a sum of squares or is far from a sum of squares. We show that if we require that the property tester always accepts YES instances and…
We prove lower bounds on the complexity of finding $\epsilon$-stationary points (points $x$ such that $\|\nabla f(x)\| \le \epsilon$) of smooth, high-dimensional, and potentially non-convex functions $f$. We consider oracle-based complexity…
We introduce a framework of layered subsets, and give a sufficient condition for when a set system supports an agreement test. Agreement testing is a certain type of property testing that generalizes PCP tests such as the plane vs. plane…
We study the fundamental problem of high-dimensional mean estimation in a robust model where a constant fraction of the samples are adversarially corrupted. Recent work gave the first polynomial time algorithms for this problem with…
Our contribution in this paper is two folded. We consider first the case of linear programming with real coefficients and give a method which allows the computation of a new upper bound on the distance from the origin to a feasible point.…
We study quantum algorithms for testing bipartiteness and expansion of bounded-degree graphs. We give quantum algorithms that solve these problems in time O(N^(1/3)), beating the Omega(sqrt(N)) classical lower bound. For testing expansion,…
We consider the problem of testing whether an unknown low-degree polynomial $p$ over $\mathbb{R}^n$ is sparse versus far from sparse, given access to noisy evaluations of the polynomial $p$ at \emph{randomly chosen points}. This is a…
We establish new hardness results for decision tree optimization problems, adding to a line of work that dates back to Hyafil and Rivest in 1976. We prove, under randomized ETH, superpolynomial lower bounds for two basic problems: given an…
We study the problem of \emph{robust satisfiability} of systems of nonlinear equations, namely, whether for a given continuous function $f:\,K\to\mathbb{R}^n$ on a~finite simplicial complex $K$ and $\alpha>0$, it holds that each function…
In this note we aim to give a new, elementary proof of a statement that was first proved by Timofte. It says that a symmetric real polynomial $F$ of degree $d$ in $n$ variables is positive on $\R^n$ (on $\R^{n}_{\geq 0}$) if and only if it…
We provide new gradient-based methods for efficiently solving a broad class of ill-conditioned optimization problems. We consider the problem of minimizing a function $f : \mathbb{R}^d \rightarrow \mathbb{R}$ which is implicitly…
Building on existing $hp$-adaptive algorithms driven by equilibrated-flux estimators from [ESAIM Math. Model. Numer. Anal. 57 (2023), 329--366] and the references therein, we propose a novel $h$-adaptive algorithm for a fixed polynomial…