Subspace-Orbit Randomized Decomposition for Low-Rank Matrix Approximations

Abstract

An efficient, accurate and reliable approximation of a matrix by one of lower rank is a fundamental task in numerical linear algebra and signal processing applications. In this paper, we introduce a new matrix decomposition approach termed Subspace-Orbit Randomized singular value decomposition (SOR-SVD), which makes use of random sampling techniques to give an approximation to a low-rank matrix. Given a large and dense data matrix of size m × n with numerical rank k, where k min{m, n}, the algorithm requires a few passes through data, and can be computed in O(mnk) floating-point operations. Moreover, the SOR-SVD algorithm can utilize advanced computer architectures, and, as a result, it can be optimized for maximum efficiency. The SOR-SVD algorithm is simple, accurate, and provably correct, and outperforms previously reported techniques in terms of accuracy and efficiency. Our numerical experiments support these claims.

Compressed Randomized Utv Decompositions for Low-rank Matrix Approximations in Data Science

“…It captures most attributes of the data by means of forming the smaller matrix through linear combinations of both rows and columns of the given matrix, and then applies the SVD for further computations. The work in [43] proposed a randomized algorithm termed subspace-orbit randomized SVD (SOR-SVD). SOR-SVD alternately projects the matrix onto its column and row space.…”

Compressed Randomized UTV Decompositions for Low-Rank Matrix Approximations

“…This, as explained later on, reduces the computational costs of CoR-UTV compared to TSR-SVD. The key difference between CoR-UTV and SOR-SVD [68], however, lies in the computation of the compressed matrix D; SOR-SVD applies a truncated SVD and, as result, gives a rankk approximation to A, while CoR-UTV employs a columnpivoted QR decomposition and returns a rank-ℓ approximation. Nevertheless, the SVD is computationally more expensive than the column-pivoted QR, and standard techniques to computing it are challenging to parallelize [45], [46], [47].…”

“…We borrow material from [68] since the two algorithms, CoR-UTV and SOR-SVD, have a few steps similar. However,…”

“…We provide theoretical guarantees on the accuracy of these approximations in terms of the Frobenius and spectral norm. To this end, we first state a theorem from [68].…”

Design of Compressed Sensing System With Probability-Based Prior Information

“…In order to find an unique mapping between the signal x and the measurement y, the constraint of sparsity on x can be utilized [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22], [23], [24], [25], [26], [27], [28], [29], [30], [31], [32], [33], [34], [35], [36], Corresponding author: Sheng Li (email: shengli@zjut.edu.cn). [37], [38], [39], [40], [41], [42], [43], [44], [45]. The sparse representation for x can be expressed as:…”