Randomized Matrix Decompositions Using <i>R</i>

Matrix decompositions are fundamental tools in the area of applied\nmathematics, statistical computing, and machine learning. In particular,\nlow-rank matrix decompositions are vital, and widely used for data analysis,\ndimensionality reduction, and data compression. Massive datasets, however, pose\na computational challenge for traditional algorithms, placing significant\nconstraints on both memory and processing power. Recently, the powerful concept\nof randomness has been introduced as a strategy to ease the computational load.\nThe essential idea of probabilistic algorithms is to employ some amount of\nrandomness in order to derive a smaller matrix from a high-dimensional data\nmatrix. The smaller matrix is then used to compute the desired low-rank\napproximation. Such algorithms are shown to be computationally efficient for\napproximating matrices with low-rank structure. We present the \\proglang{R}\npackage rsvd, and provide a tutorial introduction to randomized matrix\ndecompositions. Specifically, randomized routines for the singular value\ndecomposition, (robust) principal component analysis, interpolative\ndecomposition, and CUR decomposition are discussed. Several examples\ndemonstrate the routines, and show the computational advantage over other\nmethods implemented in R.\n