矩阵采样算法专题
Topics in Matrix Sampling Algorithms
摘要：
我们研究了线性代数的三个基本问题，这些问题是各种机器学习应用的核心，即：1）“基于低秩列的矩阵逼近”。我们给出一个矩阵A和一个目标秩k。目标是选择A的列的子集，并且仅使用这些列，计算A的秩k近似，其与使用所有列获得的秩k近似一样好; 2）“最小二乘回归中的Coreset构造”。我们给出一个矩阵A和一个向量b。考虑最小化的（过约束）最小二乘问题||ax-B||，在D中的所有向量x上。域D表示解的约束，并且可以是任意的。目标是选择A和b的行的子集，并且通过仅使用这些行，找到与通过使用所有行获得的解向量一样好的解向量; 3）“K均值聚类中的特征选择”。我们给出了一组关于大量特征描述的点。目标是选择特征的子集，并且通过仅使用该子集，获得与通过使用所有特征获得的分区一样好的点的k分区。我们提出了新的算法，所有上述三个问题。我们的研究结果可以被视为后续研究的一行工作被称为“矩阵采样算法”。[Frieze，Kanna，Vempala，1998]提出了第一个这样的算法的低秩矩阵近似问题。从那时起，这样的算法已经开发了其他几个问题，例如图稀疏化和线性方程求解。我们对这一研究领域的贡献是：（i）改进的低秩矩阵近似和回归算法（ii）新问题域（K-均值聚类）的算法。
英文摘要：
We study three fundamental problems of Linear Algebra, lying in the heart of various Machine Learning applications, namely: 1)"Low-rank Column-based Matrix Approximation". We are given a matrix A and a target rank k. The goal is to select a subset of columns of A and, by using only these columns, compute a rank k approximation to A that is as good as the rank k approximation that would have been obtained by using all the columns; 2) "Coreset Construction in Least-Squares Regression". We are given a matrix A and a vector b. Consider the (over-constrained) least-squares problem of minimizing ||Ax-b||, over all vectors x in D. The domain D represents the constraints on the solution and can be arbitrary. The goal is to select a subset of the rows of A and b and, by using only these rows, find a solution vector that is as good as the solution vector that would have been obtained by using all the rows; 3) "Feature Selection in K-means Clustering". We are given a set of points described with respect to a large number of features. The goal is to select a subset of the features and, by using only this subset, obtain a k-partition of the points that is as good as the partition that would have been obtained by using all the features. We present novel algorithms for all three problems mentioned above. Our results can be viewed as follow-up research to a line of work known as "Matrix Sampling Algorithms". [Frieze, Kanna, Vempala, 1998] presented the first such algorithm for the Low-rank Matrix Approximation problem. Since then, such algorithms have been developed for several other problems, e.g. Graph Sparsification and Linear Equation Solving. Our contributions to this line of research are: (i) improved algorithms for Low-rank Matrix Approximation and Regression (ii) algorithms for a new problem domain (K-means Clustering).
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。
