诸如奇异值分解和列主元 QR 分解等标准秩揭示分解在 GPU 上难以高效实现。其主要困难在于标准算法无法将大部分运算转化为 Level-3 BLAS 操作。本文提出了两种替代算法，用于计算形如 $A = U T V^*$ 的秩揭示分解，其中 $U$ 和 $V$ 为正交矩阵，$T$ 为三角矩阵。这两种算法均采用随机投影技术，将大部分浮点运算转化为在 GPU 上极其高效的矩阵乘法。数值实验表明，与经过精细调优的 GPU SVD 实现相比，这些算法实现了一个数量级的加速，同时其低秩近似误差接近 SVD。
英文摘要：
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 $A = U T V^*$, 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.
In this paper we introduce a new column selection strategy, named here ``Deviation Maximization", and apply it to compute rank-revealing QR factorizations as an alternative to the well known block version of the QR factorization with the column pivoting method, called QP3 and currently implemented in LAPACK's xgeqp3 routine. We show that the resulting algorithm, named QRDM, has similar rank-revealing properties of QP3 and better execution times. We present numerical test results on a wide data set of numerically singular matrices, which has become a reference in the recent literature.
We introduce a Generalized Randomized QR-decomposition that may be applied to arbitrary products of matrices and their inverses, without needing to explicitly compute the products or inverses. This factorization is a critical part of a communication-optimal spectral divide-and-conquer algorithm for the nonsymmetric eigenvalue problem. In this paper, we establish that this randomized QR-factorization satisfies the strong rank-revealing properties. We also formally prove its stability, making it suitable in applications. Finally, we present numerical experiments which demonstrate that our theore…
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…
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…
Given a matrix $A$ of size $m\times n$, the manuscript describes a algorithm for computing a QR factorization $AP=QR$ where $P$ is a permutation matrix, $Q$ is orthonormal, and $R$ is upper triangular. The algorithm is blocked, to allow it to be implemented efficiently. The need for single vector pivoting in classical algorithms for computing QR factorizations is avoided by the use of randomized sampling to find blocks of pivot vectors at once. The advantage of blocking becomes particularly pronounced when $A$ is very large, and possibly stored out-of-core, or on a distributed memory machine. …
RSVDPACK: An implementation of randomized algorithms for computing the singular value, interpolative, and CUR decompositions of matrices on multi-core and GPU architectures
University of Colorado at Boulder（科罗拉多大学博尔德分校）；University of Texas at Austin（得克萨斯大学奥斯汀分校）；Universitat Jaume I（海梅一世大学）
This paper describes efficient algorithms for computing rank-revealing factorizations of matrices that are too large to fit in RAM, and must instead be stored on slow external memory devices such as solid-state or spinning disk hard drives (out-of-core or out-of-memory). Traditional algorithms for computing rank revealing factorizations, such as the column pivoted QR factorization, or techniques for computing a full singular value decomposition of a matrix, are very communication intensive. They are naturally expressed as a sequence of matrix-vector operations, which become prohibitively expen…
In this note we describe two modifications of the ScaLAPACK subroutines PxGEQPF for computing the QR factorization with the Businger-Golub column pivoting. First, we resolve a subtle numerical instability in the same way as we have done it for the LAPACK subroutines xGEQPF, xGEQP3 in 2006. [LAPACK Working Note 176 (2006); ACM Trans. Math. Softw. 2008]. The problem originates in the first release of LINPACK in the 1970's: due to severe cancellations in the down-dating of partial column norms, the pivoting procedure may be in the dark completely about the true norms of the pivot column candidate…
We are interested in solving linear systems arising from three applications: (1) kernel methods in machine learning, (2) discretization of boundary integral equations from mathematical physics, and (3) Schur complements formed in the factorization of many large sparse matrices. The coefficient matrices are often data-sparse in the sense that their off-diagonal blocks have low numerical ranks; specifically, we focus on "hierarchically off-diagonal low-rank (HODLR)" matrices. We introduce algorithms for factorizing HODLR matrices and for applying the factorizations on a GPU. The algorithms lever…
This survey describes probabilistic algorithms for linear algebra computations, such as factorizing matrices and solving linear systems. It focuses on techniques that have a proven track record for real-world problem instances. The paper treats both the theoretical foundations of the subject and the practical computational issues. Topics covered include norm estimation; matrix approximation by sampling; structured and unstructured random embeddings; linear regression problems; low-rank approximation; subspace iteration and Krylov methods; error estimation and adaptivity; interpolatory and CUR …
