Abstract
Big data has in recent years gained ground in many scientific and engineering problems. It seems to some extent prohibitive for traditional matrix decomposition methods (i.e. QR, SVD, EVD, etc.) to handle such large-scale problems involving data matrix. Many researchers have developed several algorithms to decompose such big data matrices. An accuracy-enhanced randomized singular value decomposition method (referred to as AE-RSVDM) with orthonormalization recently becomes the state-of-the-art to factorize large data matrices with satisfactory speed and accuracy. In our paper, low-rank matrix approximations based on randomization are studied, with emphasis on accelerating the computational efficiency on large data matrices. By this, we accelerate the AE-RSVDM with modified normalized power iteration to result in an accelerated version. The accelerated version is grounded on a two-stage scheme. The first stage seeks to find the range of a sketch matrix which involves a Gaussian random matrix. A low-dimensional space is then created from the high-dimensional data matrix via power iteration. Numerical experiments on matrices of different sizes demonstrate that our accelerated variant achieves speedups while attaining the same reconstruction error as the AE-RSVDM with orthonormalization. And with data from Google art project, we have made known the computational speed-up of the accelerated variant over the AE-RSVDM algorithm for decomposing large data matrices with low-rank form.
Highlights
Matrices in many applications appear with a low-rank structure and matrix factorization approaches are often employed to develop compacted and informative observations to make data calculation and interpretation easier
Inspired by the work of Martinsson [14], we found it attractive to improve upon the speed of the AE-RSVDM
We present the so-called “accelerated AE-RSVDM with modified normalized power iteration” algorithm which is an extension of Martinsson's work
Summary
Matrices in many applications appear with a low-rank structure and matrix factorization approaches are often employed to develop compacted and informative observations to make data calculation and interpretation easier. A long-established method is the almighty singular value decomposition (SVD) which discovers the finest low-rank approximation of a data matrix. This illustration permits data analysts to analyze or work with the matrix D using the factor matrices E and F as an alternative of the full matrix, which is more efficient both computationally and memory usage. These smaller factor matrices can offer definite structures to achieve the desired result by analyzing a data matrix [1]. For a Joseph Roger Arhin et al.: An Accelerated Accuracy-enhanced Randomized Singular Value Decomposition for Factorizing Matrices with Low-rank Structure comprehensive technical outline of SVD, one can refer to the work of Trefethen et al [5]
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