关于矩阵CGR/CUR近似的新结果
New Results on CGR/CUR Approximation of a Matrix
AI总结：
本文提出加速矩阵近似计算的方法，并探讨通过预处理缩小难处理输入类别的方向，展示了在低计算成本下获得合理CUR近似的效果，同时扩展至快速多极方法和共轭梯度算法。
摘要：
CUR和低秩近似是数值线性代数中最基本的主题之一，广泛应用于现代计算的许多重要领域，从机器学习理论和神经网络到数据挖掘和分析。我们首先大幅加速了平均输入矩阵此类近似的计算，然后展示了某些狭窄的难处理输入类别，最后指出通过使用准高斯结构乘子预处理来进一步缩小此类别的潜在方向。我们对各种现实输入进行的广泛数值测试，从奇异矩阵数据库中提取的正则化数据，一致产生了合理接近的CUR近似，且计算成本低。我们的结果和方法在矩阵计算的重要主题中有多种高效应用。我们的研究提供了新的见解，并通过采样和超采样设计更快的低秩近似算法，通过无消元的高斯消元和块高斯消元，以及近似与矩阵ν最小奇异值相关的奇异空间。本文最后提出了对快速多极方法和共轭梯度算法的新型扩展。
英文摘要：
CUR and low-rank approximations are among most fundamental subjects of numerical linear algebra, with a wide range of applications to a variety of highly important areas of modern computing, which range from the machine learning theory and neural networks to data mining and analysis. We first dramatically accelerate computation of such approximations for the average input matrix, then show some narrow classes of hard inputs for our algorithms, and finally point out a tentative direction to narrowing such classes further by means of pre-processing with quasi Gaussian structured multipliers. Our extensive numerical tests with a variety of real world inputs for regularization from Singular Matrix Database have consistently produced reasonably close CUR approximations at a low computational cost. There is a variety of efficient applications of our results and our techniques to important subjects of matrix computations. Our study provides new insights and enable design of faster algorithms for low-rank approximation by means of sampling and oversampling, for Gaussian elimination with no pivoting and block Gaussian elimination, and for the approximation of railing singular spaces associated with the $ν$ smallest singular values of a matrix having numerical nullity. We conclude the paper with novel extensions of our acceleration to the Fast Multipole Method and the Conjugate Gradient Algorithms. (This report is an outline of our paper in arXiv:1611.01391.)
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。
