Related papers: New Lower Bounds for the Schur-Siegel-Smyth Trace …
Discharging arguments demonstrate a connection between local structure and global averages. This makes it an effective tool for proving lower bounds on the density of special sets in infinite grids. However, the minimum density of an…
We prove an asymptotically tight bound (asymptotic with respect to the number of polynomials for fixed degrees and number of variables) on the number of semi-algebraically connected components of the realizations of all realizable sign…
Given a graph and a subset of its nodes, referred to as source nodes, the minimum broadcast problem asks for the minimum number of steps in which a signal can be transmitted from the sources to all other nodes in the graph. In each step,…
Rank-constrained optimization problems have received an increasing intensity of interest recently, because many optimization problems in communications and signal processing applications can be cast into a rank-constrained optimization…
The framework of Integral Quadratic Constraints of Lessard et al. (2014) reduces the computation of upper bounds on the convergence rate of several optimization algorithms to semi-definite programming (SDP). Followup work by Nishihara et…
We study data-driven least squares (LS) problems with semidefinite (SD) constraints and derive finite-sample guarantees on the spectrum of their optimal solutions when these constraints are relaxed. In particular, we provide a high…
We improve the lower bound on worst case trace reconstruction from $\Omega\left(\frac{n^{5/4}}{\sqrt{\log n}}\right)$ to $\Omega\left(\frac{n^{3/2}}{\log^{7} n}\right)$. As a consequence, we improve the lower bound on average case trace…
Linear differential equations and recurrences reveal many properties about their solutions. Therefore, these equations are well-suited for representing solutions and computing with special functions. We identify a large class of existing…
An important yet challenging problem in numerical linear algebra is finding a principal submatrix with maximum determinant from a given symmetric positive semidefinite matrix. This problem arises in experimental design, statistics, and…
We derive a new estimate of the size of finite sets of points in metric spaces with few distances. The following applications are considered: (1) we improve the Ray-Chaudhuri--Wilson bound of the size of uniform intersecting families of…
A fertile area of recent research has demonstrated concrete polynomial time lower bounds for solving natural hard problems on restricted computational models. Among these problems are Satisfiability, Vertex Cover, Hamilton Path, Mod6-SAT,…
Traces of inverse powers of a positive definite symmetric tridiagonal matrix give lower bounds of the minimal singular value of an upper bidiagonal matrix. In a preceding work, a formula for the traces which gives the diagonal entries of…
The classical branch-and-bound algorithm for the integer feasibility problem has exponential worst case complexity. We prove that it is surprisingly efficient on reformulated problems, in which the columns of the constraint matrix are…
We study convergence lower bounds of without-replacement stochastic gradient descent (SGD) for solving smooth (strongly-)convex finite-sum minimization problems. Unlike most existing results focusing on final iterate lower bounds in terms…
The quadratic shortest path problem is the problem of finding a path in a directed graph such that the sum of interaction costs over all pairs of arcs on the path is minimized. We derive several semidefinite programming relaxations for the…
We consider the problem of tracking a target whose dynamics is modeled by a continuous It\=o semi-martingale. The aim is to minimize both deviation from the target and tracking efforts. We establish the existence of asymptotic lower bounds…
Assuming that there exist (infinitely many) Siegel zeros, we show that the (Rosser-)Jurkat-Richert bounds in the linear sieve cannot be improved, and similarly look at Iwaniec's lower bound on Jacobsthal's problem, as well as minor…
In the orthogonal range reporting problem, we are to preprocess a set of $n$ points with integer coordinates on a $U \times U$ grid. The goal is to support reporting all $k$ points inside an axis-aligned query rectangle. This is one of the…
We give an improved upper bound for the Gr\"unbaum--Hadwiger--Ramos problem: Let $d,n,k \in \mathbb{N}$ such that $d \geq 2^n(1+2^{k-1})$. Given $2^{n+1}$ masses on $\mathbb{R}^d$, there exist $k$ hyperplanes in $\mathbb{R}^d$ that…
We give a simplified and improved lower bound for the simplex range reporting problem. We show that given a set $P$ of $n$ points in $\mathbb{R}^d$, any data structure that uses $S(n)$ space to answer such queries must have…