使用R的随机矩阵分解
Randomized Matrix Decompositions using R
摘要：
矩阵分解是应用数学、统计计算和机器学习领域的基本工具。特别是，低秩矩阵分解是至关重要的，并广泛用于数据分析，降维和数据压缩。然而，海量数据集对传统算法提出了计算挑战，对内存和处理能力都有很大的限制。最近，随机性的强大概念已经被引入作为减轻计算负荷的策略。概率算法的基本思想是使用一定量的随机性，以便从高维数据矩阵中导出更小的矩阵。然后使用较小的矩阵来计算所需的低秩近似。这种算法被证明是计算效率低秩结构的近似矩阵。我们提出了\proglang{R}包rsvd，并提供了一个教程介绍随机矩阵分解。具体而言，随机例程的奇异值分解，（强大的）主成分分析，插值分解，CUR分解进行了讨论。几个例子演示了例程，并显示了计算优势，比其他方法在R。
英文摘要：
Matrix decompositions are fundamental tools in the area of applied mathematics, statistical computing, and machine learning. In particular, low-rank matrix decompositions are vital, and widely used for data analysis, dimensionality reduction, and data compression. Massive datasets, however, pose a computational challenge for traditional algorithms, placing significant constraints on both memory and processing power. Recently, the powerful concept of randomness has been introduced as a strategy to ease the computational load. The essential idea of probabilistic algorithms is to employ some amount of randomness in order to derive a smaller matrix from a high-dimensional data matrix. The smaller matrix is then used to compute the desired low-rank approximation. Such algorithms are shown to be computationally efficient for approximating matrices with low-rank structure. We present the \proglang{R} package rsvd, and provide a tutorial introduction to randomized matrix decompositions. Specifically, randomized routines for the singular value decomposition, (robust) principal component analysis, interpolative decomposition, and CUR decomposition are discussed. Several examples demonstrate the routines, and show the computational advantage over other methods implemented in R.
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