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We give an efficient algorithm for finding sparse approximate solutions to linear systems of equations with nonnegative coefficients. Unlike most known results for sparse recovery, we do not require {\em any} assumption on the matrix other…

We provide faster algorithms and improved sample complexities for approximating the top eigenvector of a matrix. Offline Setting: Given an $n \times d$ matrix $A$, we show how to compute an $\epsilon$ approximate top eigenvector in time…

Preconditioning has long been a staple technique in optimization, often applied to reduce the condition number of a matrix and speed up the convergence of algorithms. Although there are many popular preconditioning techniques in practice,…

Factorization of large dense matrices are ubiquitous in engineering and data science applications, e.g. preconditioners for iterative boundary integral solvers, frontal matrices in sparse multifrontal solvers, and computing the determinant…

We present a randomized algorithm that, on input a symmetric, weakly diagonally dominant n-by-n matrix A with m nonzero entries and an n-vector b, produces a y such that $\norm{y - \pinv{A} b}_{A} \leq \epsilon \norm{\pinv{A} b}_{A}$ in…

Given a multiset $S$ of $n$ positive integers and a target integer $t$, the Subset Sum problem asks to determine whether there exists a subset of $S$ that sums up to $t$. The current best deterministic algorithm, by Koiliaris and Xu…

We propose a rank-one Riemannian subspace descent algorithm for computing symmetric positive definite (SPD) solutions to nonlinear matrix equations arising in control theory, dynamic programming, and stochastic filtering. For solution…

Motivated by applications in recommender systems, web search, social choice and crowdsourcing, we consider the problem of identifying the set of top $K$ items from noisy pairwise comparisons. In our setting, we are non-actively given $r$…

We develop a novel unified randomized block-coordinate primal-dual algorithm to solve a class of nonsmooth constrained convex optimization problems, which covers different existing variants and model settings from the literature. We prove…

There has been a long history of using ordinary differential equations (ODEs) to understand the dynamics of discrete-time algorithms (DTAs). Surprisingly, there are still two fundamental and unanswered questions: (i) it is unclear how to…

There is a recent interest on first-order methods for linear programming (LP). In this paper,we propose a stochastic algorithm using variance reduction and restarts for solving sharp primal-dual problems such as LP. We show that the…

We study quantum algorithms for approximating Lasserre's hierarchy values for polynomial optimization. Let $f,g_1,\ldots,g_m$ be real polynomials in $n$ variables and $f^\star$ the infimum of $f$ over the semialgebraic set $S(g)=\{x:…

Min-plus product of two $n\times n$ matrices is a fundamental problem in algorithm research. It is known to be equivalent to APSP, and in general it has no truly subcubic algorithms. In this paper, we focus on the min-plus product on a…

We present a polynomial-time reduction from solving noisy linear equations over $\mathbb{Z}/q\mathbb{Z}$ in dimension $\Theta(k\log n/\mathsf{poly}(\log k,\log q,\log\log n))$ with a uniformly random coefficient matrix to noisy linear…

We describe a probabilistic, {\it sublinear} runtime, measurement-optimal system for model-based sparse recovery problems through dimensionality reducing, {\em dense} random matrices. Specifically, we obtain a linear sketch $u\in \R^M$ of a…

The quest for an algorithm that solves an $n\times n$ linear system in $O(n^2)$ time complexity, or $O(n^2 \text{poly}(1/\epsilon))$ when solving up to $\epsilon$ relative error, is a long-standing open problem in numerical linear algebra…

We consider the problem of sampling from a log-concave distribution $\pi(\theta) \propto e^{-f(\theta)}$ constrained to a polytope $K:=\{\theta \in \mathbb{R}^d: A\theta \leq b\}$, where $A\in \mathbb{R}^{m\times d}$ and $b \in…

Gradient-based minimax optimal algorithms have greatly promoted the development of continuous optimization and machine learning. One seminal work due to Yurii Nesterov [Nes83a] established $\tilde{\mathcal{O}}(\sqrt{L/\mu})$ gradient…

While quantum algorithms for solving large scale systems of linear equations offer potentially exponential speedups, their application has largely been confined to sparse matrices. This work extends the scope of these algorithms to a broad…

Packing and covering linear programs belong to the narrow class of linear programs that are efficiently solvable in parallel and distributed models of computation, yet are a powerful modeling tool for a wide range of fundamental problems in…