The QLP decomposition is one of the effective algorithms to approximate singular value decomposition (SVD) in numerical linear algebra. In this paper, we propose some single-pass randomized QLP decomposition algorithms for computing the low-rank matrix approximation. Compared with the deterministic QLP decomposition, the complexity of the proposed algorithms does not increase significantly and the system matrix needs to be accessed only once. Therefore, our algorithms are very suitable for a large matrix stored outside of memory or generated by stream data. In the error analysis, we give the bounds of matrix approximation error and singular value approximation error. Numerical experiments also reported to verify our results.
In this paper, we introduce a randomized QLP decomposition called Rand-QLP. Operating on a matrix $\bf A$, Rand-QLP gives ${\bf A}={\bf QLP}^T$, where $\bf Q$ and $\bf P$ are orthonormal, and $\bf L$ is lower-triangular. Under the assumption that the rank of the input matrix is $k$, we derive several error bounds for Rand-QLP: bounds for the first $k$ approximate singular values and for the trailing block of the middle factor $\bf L$, which show that the decomposition is rank-revealing; bounds for the distance between approximate subspaces and the exact ones for all four fundamental subspaces …
We present Flip-Flop Spectrum-Revealing QR (Flip-Flop SRQR) factorization, a significantly faster and more reliable variant of the QLP factorization of Stewart, for low-rank matrix approximations. Flip-Flop SRQR uses SRQR factorization to initialize a partial column pivoted QR factorization and then compute a partial LQ factorization. As observed by Stewart in his original QLP work, Flip-Flop SRQR tracks the exact singular values with "considerable fidelity". We develop singular value lower bounds and residual error upper bounds for Flip-Flop SRQR factorization. In situations where singular va…
Many applications in scientific computing and data science require the computation of a rank-revealing factorization of a large matrix. In many of these instances the classical algorithms for computing the singular value decomposition are prohibitively computationally expensive. The randomized singular value decomposition can often be helpful, but is not effective unless the numerical rank of the matrix is substantially smaller than the dimensions of the matrix. We introduce a new randomized algorithm for producing rank-revealing factorizations based on existing work by Demmel, Dumitriu and Ho…
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 SVD and 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 reorthogonalizat…
Pass-efficient Randomized Algorithms for Low-rank Approximation of Quaternion Matrices
Innopolis University（伊诺波利斯大学）；Research Center of the Artificial Intelligence Institute, and Lab of Machine Learning and Knowledge Representation, Innopolis University（伊诺波利斯大学人工智能研究所研究中心及机器学习与知识表示实验室）
Based on the column pivoted QR decomposition, we propose some randomized algorithms including pass-efficient ones for the generalized CUR decompositions of matrix pair and matrix triplet. Detailed error analyses of these algorithms are provided. Numerical experiments are given to test the proposed randomized algorithms.
Big data analysis has become a crucial part of new emerging technologies such as the internet of things, cyber-physical analysis, deep learning, anomaly detection, etc. Among many other techniques, dimensionality reduction plays a key role in such analyses and facilitates feature selection and feature extraction. Randomized algorithms are efficient tools for handling big data tensors. They accelerate decomposing large-scale data tensors by reducing the computational complexity of deterministic algorithms and the communication among different levels of the memory hierarchy, which is the main bo…
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…
In this work, a novel rank-revealing matrix decomposition algorithm termed Compressed Randomized UTV (CoR-UTV) decomposition along with a CoR-UTV variant aided by the power method technique is proposed. CoR-UTV computes an approximation to a low-rank input matrix by making use of random sampling schemes. Given a large and dense matrix of size $m\times n$ with numerical rank $k$, where $k \ll \text{min} \{m,n\}$, CoR-UTV requires a few passes over the data, and runs in $O(mnk)$ floating-point operations. Furthermore, CoR-UTV can exploit modern computational platforms and can be optimized for ma…
Department of Computer Science（计算机科学系）；Tsinghua National Lab of Information Science and Technology（清华大学信息科学与技术国家实验室）；Tsinghua University（清华大学）；Institute for Interdisciplinary Information Sciences（交叉信息科学与工程研究院）；Department of Electronic Engineering（电子工程系）；Institute of Computing Technology（计算技术研究所）；Chinese Academy of Sciences（中国科学院）；Old Dominion University（老 Dominion 大学）
Cornell University（康奈尔大学）；University of Wisconsin-Madison（威斯康星大学麦迪逊分校）；Columbia University（哥伦比亚大学）；California Institute of Technology（加州理工学院）
This paper describes a new algorithm for computing a low-Tucker-rank approximation of a tensor. The method applies a randomized linear map to the tensor to obtain a sketch that captures the important directions within each mode, as well as the interactions among the modes. The sketch can be extracted from streaming or distributed data or with a single pass over the tensor, and it uses storage proportional to the degrees of freedom in the output Tucker approximation. The algorithm does not require a second pass over the tensor, although it can exploit another view to compute a superior approxim…
