Randomized Algorithms for Computing Low-Rank Matrix Approximation

Low-rank matrix approximation is extremely useful in the analysis of data that arises in scientific computing, engineering applications, and data science. However, as data sizes grow, traditional low-rank matrix approximation methods, such as singular value decomposition (SVD) and column pivoting QR decomposition (CPQR), are either prohibitively expensive or cannot provide sufficiently accurate results. A solution is to use randomized low-rank matrix approximation methods such as randomized SVD and randomized LU decomposition on extremely large data sets. In this dissertation, we focus on the randomized LU decomposition method. First, we employ a reorthogonalization procedure to perform the power iteration of the existing randomized LU algorithm to compensate for the rounding errors caused by the power method. Then to solve the fixed precision low rank approximation problem, we block the existing randomized LU algorithm. Our proposed randomized blocked LU algorithm is accurate and has comparable speed with randomized blocked QB algorithm by Martinsson and Voronin. Then we propose a novel randomized LU algorithm, called PowerLU, for the fixed low-rank approximation problem. PowerLU allows for an arbitrary number of passes of the input matrix, $v \geq 2$. Recall that the existing randomized LU decomposition only allows an even number of passes. We prove the theoretical relationship between PowerLU and the existing randomized LU. Numerical experiments show that our proposed PowerLU is generally faster than the existing randomized LU decomposition, while remaining accurate. We also propose a version of PowerLU, called PowerLU_FP, for the fixed precision low-rank matrix approximation problem. PowerLU_FP is based on an efficient blocked adaptive rank determination Algorithm 18 proposed in this dissertation. We present numerical experiments that show that PowerLU_FP can achieve almost the same accuracy and is faster than the randomized blocked QB algorithm. We finally propose a single-pass algorithm based on LU factorization. Tests show that the accuracy of our single-pass algorithm is comparable with the existing single-pass algorithms.