Related papers: Fast algorithms for differential equations in posi…
In this paper, an exact algorithm in polynomial time is developed to solve unrestricted binary quadratic programs. The computational complexity is $O\left( n^{\frac{15}{2}}\right) $, although very conservative, it is sufficient to prove…
Positive linear programs (LP), also known as packing and covering linear programs, are an important class of problems that bridges computer science, operations research, and optimization. Despite the consistent efforts on this problem, all…
Linear differential equations are ubiquitous in science and engineering. Quantum computers can simulate quantum systems, which are described by a restricted type of linear differential equations. Here we extend quantum simulation algorithms…
We study the limitations and fast-forwarding of quantum algorithms for linear ordinary differential equation (ODE) systems with a particular focus on non-quantum dynamics, where the coefficient matrix in the ODE is not anti-Hermitian or the…
Assuming the Generalized Riemann Hypothesis, we design a deterministic algorithm that, given a prime p and positive integer m=o(sqrt(p)/(log p)^4), outputs an elliptic curve E over the finite field F_p for which the cardinality of E(F_p) is…
This paper presents fast first-order methods for solving linear programs (LPs) approximately. We adapt online linear programming algorithms to offline LPs and obtain algorithms that avoid any matrix multiplication. We also introduce a…
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…
Sorting has a natural generalization where the input consists of: (1) a ground set $X$ of size $n$, (2) a partial oracle $O_P$ specifying some fixed partial order $P$ on $X$ and (3) a linear oracle $O_L$ specifying a linear order $L$ that…
We study the problem of estimating the covariance matrix of a high-dimensional distribution when a small constant fraction of the samples can be arbitrarily corrupted. Recent work gave the first polynomial time algorithms for this problem…
We consider the problem of Robust PCA in the fully and partially observed settings. Without corruptions, this is the well-known matrix completion problem. From a statistical standpoint this problem has been recently well-studied, and…
We present an efficient, nearly optimal quantum algorithm for solving linear matrix differential equations, with applications to the simulation of open quantum systems and beyond. For unitary or dissipative dynamics, the algorithm computes…
In this paper, we develop fast procedures for solving linear systems arising from discretization of ordinary and partial differential equations with Caputo fractional derivative w.r.t time variable. First, we consider a finite difference…
We establish new exponential in dimension lower bounds for the Maximum Halfspace Discrepancy problem, which models linear classification. Both are fundamental problems in computational geometry and machine learning in their exact and…
We consider supervised learning problems within the positive-definite kernel framework, such as kernel ridge regression, kernel logistic regression or the support vector machine. With kernels leading to infinite-dimensional feature spaces,…
We reveal a complexity chasm, separating the trinomial and tetranomial cases, for solving univariate sparse polynomial equations over certain local fields. First, for any fixed field $K\in\{\mathbb{Q}_2,\mathbb{Q}_3,\mathbb{Q}_5,\ldots\}$,…
In this work we revisit the elementary scheduling problem $1||\sum p_j U_j$. The goal is to select, among $n$ jobs with processing times and due dates, a subset of jobs with maximum total processing time that can be scheduled in sequence…
Solving linear systems and computing eigenvalues are two fundamental problems in linear algebra. For solving linear systems, many efficient quantum algorithms have been discovered. For computing eigenvalues, currently, we have efficient…
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 develop the first quantum algorithm for the constrained portfolio optimization problem. The algorithm has running time $\widetilde{O} \left( n\sqrt{r} \frac{\zeta \kappa}{\delta^2} \log \left(1/\epsilon\right) \right)$, where $r$ is the…
We present an algorithm that given a linear program with $n$ variables, $m$ constraints, and constraint matrix $A$, computes an $\epsilon$-approximate solution in $\tilde{O}(\sqrt{rank(A)}\log(1/\epsilon))$ iterations with high probability.…