The PowerURV algorithm for computing rank-revealing full factorizations

Gopal, Abinand; Martinsson, Per-Gunnar

doi:10.48550/arxiv.1812.06007

Cited by 3 publications

(3 citation statements)

References 10 publications

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“…The first algorithm powerURV, discussed in Section 3, was first introduced in the technical report. 37 powerURV is built on another randomized RRUTV algorithm developed by Demmel et al in Reference 38, adding better rank revelation at a tolerable increase in computational cost. The algorithm itself is quite simple, capable of description with just a few lines of code.…”

Section: Proposed Algorithmsmentioning

confidence: 99%

Efficient algorithms for computing rank‐revealing factorizations on a GPU

Heavner

Chen

Gopal

et al. 2023

Numerical Linear Algebra App

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Standard rank‐revealing factorizations such as the singular value decomposition (SVD) and column pivoted QR factorization are challenging to implement efficiently on a GPU. A major difficulty in this regard is the inability of standard algorithms to cast most operations in terms of the Level‐3 BLAS. This article presents two alternative algorithms for computing a rank‐revealing factorization of the form , where and are orthogonal and is trapezoidal (or triangular if is square). Both algorithms use randomized projection techniques to cast most of the flops in terms of matrix‐matrix multiplication, which is exceptionally efficient on the GPU. Numerical experiments illustrate that these algorithms achieve significant acceleration over finely tuned GPU implementations of the SVD while providing low rank approximation errors close to that of the SVD.

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Section: Proposed Algorithmsmentioning

confidence: 99%

Efficient algorithms for computing rank‐revealing factorizations on a GPU

Heavner

Chen

Gopal

et al. 2023

Numerical Linear Algebra App

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show abstract

“…To test the accuracy of our methods, we randomly generate three types of the matrices based on the singular value decay speed as follows [13,38]:…”

Section: Comparison Of Accuracymentioning

confidence: 99%

Pass-Efficient Randomized LU Algorithms for Computing Low-Rank Matrix Approximation

Zhang¹,

Mascagni²

2020

Preprint

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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 paper, 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 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 ≥ 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 4.1 proposed in this paper. We present numerical experiments that show that PowerLU FP can achieve almost the same accuracy and is faster than the randomized blocked QB algorithm by Martinsson and Voronin. 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.

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“…The first algorithm POWERURV, discussed in Section 3, was first introduced in the tech report [23]. POWERURV is built on another randomized RRUTV algorithm developed by Demmel et al in [15], adding better rank revelation at a tolerable increase in computational cost.…”

Section: Introductionmentioning

confidence: 99%

Efficient algorithms for computing rank-revealing factorizations on a GPU

Heavner¹,

Chen²,

Gopal³

et al. 2021

Preprint

Self Cite

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Standard rank-revealing factorizations such as the singular value decomposition and column pivoted QR factorization are challenging to implement efficiently on a GPU. A major difficulty in this regard is the inability of standard algorithms to cast most operations in terms of the Level-3 BLAS. This paper presents two alternative algorithms for computing a rank-revealing factorization of the form A = UTV * , where U and V are orthogonal and T is triangular. Both algorithms use randomized projection techniques to cast most of the flops in terms of matrix-matrix multiplication, which is exceptionally efficient on the GPU. Numerical experiments illustrate that these algorithms achieve an order of magnitude acceleration over finely tuned GPU implementations of the SVD while providing low rank approximation errors close to that of the SVD.

show abstract

The PowerURV algorithm for computing rank-revealing full factorizations

Cited by 3 publications

References 10 publications

Efficient algorithms for computing rank‐revealing factorizations on a GPU

Efficient algorithms for computing rank‐revealing factorizations on a GPU

Pass-Efficient Randomized LU Algorithms for Computing Low-Rank Matrix Approximation

Efficient algorithms for computing rank-revealing factorizations on a GPU

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