12,522 research outputs found
Efficient Randomized Algorithms for the Fixed-Precision Low-Rank Matrix Approximation
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. Firstly, a mechanism for
calculating the approximation error in Frobenius norm is proposed, which
enables efficient adaptive rank determination for 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 P.-G.
Martinsson and S. 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-pass algorithm for calculating leading singular values, under
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-pass algorithm derived by randQB FP is much more
accurate than an existing single-pass algorithm. 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.Comment: 21 pages, 10 figure
Randomized Dynamic Mode Decomposition
This paper presents a randomized algorithm for computing the near-optimal
low-rank dynamic mode decomposition (DMD). Randomized algorithms are emerging
techniques to compute low-rank matrix approximations at a fraction of the cost
of deterministic algorithms, easing the computational challenges arising in the
area of `big data'. The idea is to derive a small matrix from the
high-dimensional data, which is then used to efficiently compute the dynamic
modes and eigenvalues. The algorithm is presented in a modular probabilistic
framework, and the approximation quality can be controlled via oversampling and
power iterations. The effectiveness of the resulting randomized DMD algorithm
is demonstrated on several benchmark examples of increasing complexity,
providing an accurate and efficient approach to extract spatiotemporal coherent
structures from big data in a framework that scales with the intrinsic rank of
the data, rather than the ambient measurement dimension. For this work we
assume that the dynamics of the problem under consideration is evolving on a
low-dimensional subspace that is well characterized by a fast decaying singular
value spectrum
Finding Structure with Randomness: Probabilistic Algorithms for Constructing Approximate Matrix Decompositions
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 × n matrix. (i) For a dense input matrix, randomized algorithms require O(mn log(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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