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We provide the first deterministic distributed synchronizer with near-optimal time complexity and message complexity overheads. Concretely, given any distributed algorithm $\mathcal{A}$ that has time complexity $T$ and message complexity…
In this paper we provide an $\tilde{O}(nd+d^{3})$ time randomized algorithm for solving linear programs with $d$ variables and $n$ constraints with high probability. To obtain this result we provide a robust, primal-dual…
We study the complexity of optimizing nonsmooth nonconvex Lipschitz functions by producing $(\delta,\epsilon)$-stationary points. Several recent works have presented randomized algorithms that produce such points using $\tilde…
We study the problem of optimal subset selection from a set of correlated random variables. In particular, we consider the associated combinatorial optimization problem of maximizing the determinant of a symmetric positive definite matrix…
We present a randomized primal-dual algorithm that solves the problem $\min_{x} \max_{y} y^\top A x$ to additive error $\epsilon$ in time $\mathrm{nnz}(A) + \sqrt{\mathrm{nnz}(A)n}/\epsilon$, for matrix $A$ with larger dimension $n$ and…
Given an $n$ by $n$ matrix $A$ and an $n$-vector $b$, along with a rational function $R(z) := D(z )^{-1} N(z)$, we show how to find the optimal approximation to $R(A) b$ from the Krylov space, $\mbox{span}( b, Ab, \ldots , A^{k-1} b)$,…
Given a set $S$ of $m$ point sites in a simple polygon $P$ of $n$ vertices, we consider the problem of computing the geodesic farthest-point Voronoi diagram for $S$ in $P$. It is known that the problem has an $\Omega(n+m\log m)$ time lower…
We provide faster randomized algorithms for computing an $\epsilon$-optimal policy in a discounted Markov decision process with $A_{\text{tot}}$-state-action pairs, bounded rewards, and discount factor $\gamma$. We provide an…
Motivated by applications in single-cell biology and metagenomics, we investigate the problem of matrix reordering based on a noisy disordered monotone Toeplitz matrix model. We establish the fundamental statistical limit for this problem…
We consider fast algorithms for monotone submodular maximization subject to a matroid constraint. We assume that the matroid is given as input in an explicit form, and the goal is to obtain the best possible running times for important…
The metric $k$-median problem is a textbook clustering problem. As input, we are given a metric space $V$ of size $n$ and an integer $k$, and our task is to find a subset $S \subseteq V$ of at most $k$ `centers' that minimizes the total…
We give an algorithm that generates a uniformly random contingency table with specified marginals, i.e. a matrix with non-negative integer values and specified row and column sums. Such algorithms are useful in statistics and combinatorics.…
This note demonstrates that we can stably recover all symmetric Toeplitz matrices $\pmb{X}_0\in\mathbb{R}^{n\times n}$ of rank at most $r$ from a number of rank-one subgaussian measurements on the order of $r\log^{2} n$ with an…
The authors recently gave an $n^{O(\log\log n)}$ time membership query algorithm for properly learning decision trees under the uniform distribution (Blanc et al., 2021). The previous fastest algorithm for this problem ran in $n^{O(\log…
We give a stochastic optimization algorithm that solves a dense $n\times n$ real-valued linear system $Ax=b$, returning $\tilde x$ such that $\|A\tilde x-b\|\leq \epsilon\|b\|$ in time: $$\tilde O((n^2+nk^{\omega-1})\log1/\epsilon),$$ where…
We introduce a novel algorithm for approximating the logarithm of the determinant of a symmetric positive definite (SPD) matrix. The algorithm is randomized and approximates the traces of a small number of matrix powers of a specially…
Matrix completion is the problem of recovering a low rank matrix by observing a small fraction of its entries. A series of recent works [KOM12,JNS13,HW14] have proposed fast non-convex optimization based iterative algorithms to solve this…
We consider $\ell_1$-Rank-$r$ Approximation over GF(2), where for a binary $m\times n$ matrix ${\bf A}$ and a positive integer $r$, one seeks a binary matrix ${\bf B}$ of rank at most $r$, minimizing the column-sum norm $||{\bf A} -{\bf…
Given a set $Z$ of $n$ positive integers and a target value $t$, the Subset Sum problem asks whether any subset of $Z$ sums to $t$. A textbook pseudopolynomial time algorithm by Bellman from 1957 solves Subset Sum in time $O(nt)$. This has…
In this paper we provide faster algorithms for approximately solving discounted Markov Decision Processes in multiple parameter regimes. Given a discounted Markov Decision Process (DMDP) with $|S|$ states, $|A|$ actions, discount factor…