使用结构化随机矩阵的更快线性代数算法
Faster Linear Algebra Algorithms with Structured Random Matrices
AI总结：
本文提出遗忘子空间注入（OSI）性质以统一分析结构化随机降维，证明其支撑低秩近似与最小二乘回归，并给出多类OSI矩阵及接近最优的加速效果。
摘要：
为了实现尽可能高的速度，从业者经常使用结构化降维映射来实现用于低秩近似和最小二乘回归的随机算法。尽管已有大量研究投入，但关于采用结构化随机矩阵的随机线性代数算法的设计与分析，仍存在一些基本问题。本文基于遗忘子空间注入（oblivious subspace injection，OSI）性质，提出了一种关于结构化降维的新视角。OSI性质是对随机矩阵的一个相对较弱的假设：当该矩阵平均保持向量长度，并且以高概率不会湮灭低维子空间中的任何向量时，该性质成立。借助OSI这一抽象，随机线性代数算法的分析可分解为两部分：（i）证明算法在使用OSI实现时有效；以及（ii）证明给定随机矩阵模型具有OSI性质。本文推进了该方案的两个部分。首先，在OSI假设下分析了用于低秩近似和最小二乘回归的标准随机算法。其次，识别出许多OSI示例，包括随机稀疏矩阵、随机三角变换以及具有张量积结构的随机矩阵。这些理论结果意味着若干基础线性代数任务可获得更快、接近最优的运行时间。本文还提供了实现指导，并给出实证证据，表明结构化随机矩阵在一系列合成问题和当代科学应用中表现出色。
英文摘要：
To achieve the greatest possible speed, practitioners regularly implement randomized algorithms for low-rank approximation and least-squares regression with structured dimension reduction maps. Despite significant research effort, basic questions remain about the design and analysis of randomized linear algebra algorithms that employ structured random matrices. This paper develops a new perspective on structured dimension reduction, based on the oblivious subspace injection (OSI) property. The OSI property is a relatively weak assumption on a random matrix that holds when the matrix preserves the length of vectors on average and, with high probability, does not annihilate any vector in a low-dimensional subspace. With the OSI abstraction, the analysis of a randomized linear algebra algorithm factors into two parts: (i) proving that the algorithm works when implemented with an OSI; and (ii) proving that a given random matrix model has the OSI property. This paper develops both parts of the program. First, it analyzes standard randomized algorithms for low-rank approximation and least-squares regression under the OSI assumption. Second, it identifies many examples of OSIs, including random sparse matrices, randomized trigonometric transforms, and random matrices with tensor product structure. These theoretical results imply faster, near-optimal runtimes for several fundamental linear algebra tasks. The paper also provides guidance on implementation, along with empirical evidence that structured random matrices offer exemplary performance for a range of synthetic problems and contemporary scientific applications.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
