Randomized algorithms for low-rank matrix approximation are investigated, with the emphasis on the fixed-precision problem and computational efficiency for handling large matrices. The algorithms are based on the so-c...
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Randomized algorithms for low-rank matrix approximation are investigated, with the emphasis on the fixed-precision problem and computational efficiency for handling large matrices. The algorithms are based on the so-called QB factorization, where Q is an orthonormal matrix. First, a mechanism for calculating the approximation error in the Frobenius norm is proposed, which enables efficient adaptive rank determination for a large and/or sparse matrix. It can be combined with any QB-form factorization algorithm in which B's rows are incrementally generated. Based on the blocked randQB algorithm by Martinsson and Voronin, this results in an algorithm called randQB_EI. Then, we further revise the algorithm to obtain a pass-efficient algorithm, randQB_FP, which is mathematically equivalent to the existing randQB algorithms and also suitable for the fixed-precision problem. Especially, randQB_FP can serve as a single-passalgorithm for calculating leading singular values, under a certain condition. With large and/or sparse test matrices, we have empirically validated the merits of the proposed techniques, which exhibit remarkable speedup and memory saving over the blocked randQB algorithm. We have also demonstrated that the single-passalgorithm derived by randQB_FP is much more accurate than an existing single-passalgorithm. And with data from a scenic image and an information retrieval application, we have shown the advantages of the proposed algorithms over the adaptive range finder algorithm for solving the fixed-precision problem.
Low-rank matrix approximations, such as the truncated singular value decomposition and the rank-revealing QR decomposition, play a central role in data analysis and scientific computing. This work surveys and extends ...
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Low-rank matrix approximations, such as the truncated singular value decomposition and the rank-revealing QR decomposition, play a central role in data analysis and scientific computing. This work surveys and extends recent research which demonstrates that randomization offers a powerful tool for performing low-rank matrix approximation. These techniques exploit modern computational architectures more fully than classical methods and open the possibility of dealing with truly massive data sets. This paper presents a modular framework for constructing randomized algorithms that compute partial matrix decompositions. These methods use random sampling to identify a subspace that captures most of the action of a matrix. The input matrix is then compressed-either explicitly or implicitly-to this subspace, and the reduced matrix is manipulated deterministically to obtain the desired low-rank factorization. In many cases, this approach beats its classical competitors in terms of accuracy, robustness, and/or speed. These claims are supported by extensive numerical experiments and a detailed error analysis. The specific benefits of randomized techniques depend on the computational environment. Consider the model problem of finding the k dominant components of the singular value decomposition of an m x n matrix. (i) For a dense input matrix, randomized algorithms require O(mnlog(k)) floating-point operations (flops) in contrast to O(mnk) for classical algorithms. (ii) For a sparse input matrix, the flop count matches classical Krylov subspace methods, but the randomized approach is more robust and can easily be reorganized to exploit multiprocessor architectures. (iii) For a matrix that is too large to fit in fast memory, the randomized techniques require only a constant number of passes over the data, as opposed to O(k) passes for classical algorithms. In fact, it is sometimes possible to perform matrix approximation with a single pass over the data.
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