Related papers: Nearly Optimal Attention Coresets
Let $\mathcal{K}$ be a discrete valued field with finite residue field. In analogy with orthogonality in the Euclidean space $\mathbb{R}^n$, there is a well-studied notion of "ultrametric orthogonality" in $\mathcal{K}^n$. In this paper,…
A graph is called a $(k,\rho)$-graph iff every node can reach $\rho$ of its nearest neighbors in at most k hops. This property proved useful in the analysis and design of parallel shortest-path algorithms. Any graph can be transformed into…
Quantum key distribution (QKD) could be the most significant application of quantum information theory. In nearly four decades, although substantial QKD protocols are developed, the BB84 protocol and its variants are still the most…
We develop and analyze a method to reduce the size of a very large set of data points in a high dimensional Euclidean space R d to a small set of weighted points such that the result of a predetermined data analysis task on the reduced set…
In this work, we study the classic submodular maximization problem under knapsack constraints and beyond. We first present an $(7/16-\varepsilon)$-approximate algorithm for single knapsack constraint, which requires…
Given a set of $n$ vectors in $\mathbb{R}^d$, the goal of the \emph{determinant maximization} problem is to pick $k$ vectors with the maximum volume. Determinant maximization is the MAP-inference task for determinantal point processes (DPP)…
We show how to find all $k$ marked elements in a list of size $N$ using the optimal number $O(\sqrt{N k})$ of quantum queries and only a polylogarithmic overhead in the gate complexity, in the setting where one has a small quantum memory.…
The study of Dense-$3$-Subhypergraph problem was initiated in Chlamt{\'{a}}c et al. [Approx'16]. The input is a universe $U$ and collection ${\cal S}$ of subsets of $U$, each of size $3$, and a number $k$. The goal is to choose a set $W$ of…
Clustering is the task of partitioning a given set of geometric objects. This is thoroughly studied when the objects are points in the euclidean space. There are also several approaches for points in general metric spaces. In this thesis we…
In the $\ell_p$-subspace sketch problem, we are given an $n\times d$ matrix $A$ with $n>d$, and asked to build a small memory data structure $Q(A,\epsilon)$ so that, for any query vector $x\in\mathbb{R}^d$, we can output a number in…
Specific data compression techniques, formalized by the concept of coresets, proved to be powerful for many optimization problems. In fact, while tightly controlling the approximation error, coresets may lead to significant speed up of the…
We study near-critical behavior in the configuration model. Let $D_n$ be the degree of a random vertex. We let $\nu_n={\mathbb E} [D_n(D_n-1)]/{\mathbb E}[D_n]$ and, assuming that $\nu_n \to 1$ as $n \to \infty$, we write…
We investigate the maximal size of an increasing subset among points randomly sampled from certain probability densities. Kerov and Vershik's celebrated result states that the largest increasing subset among $N$ uniformly random points on…
``Composable core-sets'' are an efficient framework for solving optimization problems in massive data models. In this work, we consider efficient construction of composable core-sets for the determinant maximization problem. This can also…
We prove tight lower bounds for the following variant of the counting problem considered by Aaronson, Kothari, Kretschmer, and Thaler (2020). The task is to distinguish whether an input set $x\subseteq [n]$ has size either $k$ or…
We study the amplitude-constrained additive white Gaussian noise (AWGN) channel from the perspective of near-optimal input distributions. While it is known that the capacity-achieving input is discrete with finitely many mass points, the…
We present a novel approximation algorithm for $k$-median that achieves an approximation guarantee of $1+\sqrt{3}+\epsilon$, improving upon the decade-old ratio of $3+\epsilon$. Our approach is based on two components, each of which, we…
We study the quadratically regularized optimal transport (QOT) problem for quadratic cost and compactly supported marginals $\mu$ and $\nu$. It has been empirically observed that the optimal coupling $\pi_\epsilon$ for the QOT problem has…
In the polytope membership problem, a convex polytope $K$ in $R^d$ is given, and the objective is to preprocess $K$ into a data structure so that, given a query point $q \in R^d$, it is possible to determine efficiently whether $q \in K$.…
We study the complexity of approximating the vertex expansion of graphs $G = (V,E)$, defined as \[ \Phi^V := \min_{S \subset V} n \cdot \frac{|N(S)|}{|S| |V \backslash S|}. \] We give a simple polynomial-time algorithm for finding a subset…