Randomized singular value decomposition (RSVD) is by now a well-established technique for efficiently computing an approximate singular value decomposition of a matrix. Building on the ideas that underpin RSVD, the recently proposed algorithm “randUTV” computes a full factorization of a given matrix that provides low-rank approximations with near-optimal error. Because the bulk of randUTV is cast in terms of communication-efficient operations such as matrix-matrix multiplication and unpivoted QR factorizations, it is faster than competing rank-revealing factorization methods such as column-pivoted QR in most high-performance computational settings. In this article, optimized randUTV implementations are presented for both shared-memory and distributed-memory computing environments. For shared memory, randUTV is redesigned in terms of an algorithm-by-blocks that, together with a runtime task scheduler, eliminates bottlenecks from data synchronization points to achieve acceleration over the standard blocked algorithm based on a purely fork-join approach. The distributed-memory implementation is based on the ScaLAPACK library. The performance of our new codes compares favorably with competing factorizations available on both shared-memory and distributed-memory architectures.
more »
« less
Computing rank‐revealing factorizations of matrices stored out‐of‐core
Summary This paper describes efficient algorithms for computing rank‐revealing factorizations of matrices that are too large to fit in main memory (RAM), and must instead be stored on slow external memory devices such as disks (out‐of‐core or out‐of‐memory). Traditional algorithms for computing rank‐revealing factorizations (such as the column pivoted QR factorization and the singular value decomposition) are very communication intensive as they require many vector‐vector and matrix‐vector operations, which become prohibitively expensive when data is not in RAM. Randomization allows to reformulate new methods so that large contiguous blocks of the matrix are processed in bulk. The paper describes two distinct methods. The first is a blocked version of column pivoted Householder QR, organized as a “left‐looking” method to minimize the number of the expensive write operations. The second method results employs a UTV factorization. It is organized as an algorithm‐by‐blocks to overlap computations and I/O operations. As it incorporates power iterations, it is much better at revealing the numerical rank. Numerical experiments on several computers demonstrate that the new algorithms are almost as fast when processing data stored on slow memory devices as traditional algorithms are for data stored in RAM.
more »
« less
- Award ID(s):
- 1952735
- NSF-PAR ID:
- 10454845
- Date Published:
- Journal Name:
- Concurrency and Computation: Practice and Experience
- ISSN:
- 1532-0626
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
More Like this
-
-
The use of batched matrix computations recently gained a lot of interest for applications, where the same operation is applied to many small independent matrices. The batched computational pattern is frequently encountered in applications of data analytics, direct/iterative solvers and preconditioners, computer vision, astrophysics, and more, and often requires specific designs for vectorization and extreme parallelism to map well on today's high-end many-core architectures. This has led to the development of optimized software for batch computations, and to an ongoing community effort to develop standard interfaces for batched linear algebra software. Furthering these developments, we present GPU design and optimization techniques for high-performance batched one-sided factorizations of millions of tiny matrices (of size 32 and less). We quantify the effects and relevance of different techniques in order to select the best-performing LU, QR, and Cholesky factorization designs. While we adapt common optimization techniques, such as optimal memory traffic, register blocking, and concurrency control, we also show that a different mindset and techniques are needed when matrices are tiny, and in particular, sub-vector/warp in size. The proposed routines are part of the MAGMA library and deliver significant speedups compared to their counterparts in currently available vendor-optimized libraries. Notably, we tune the developments for the newest V100 GPU from NVIDIA to show speedups of up to 11.8×.more » « less
-
The matrix completion problem seeks to recover a $d\times d$ ground truth matrix of low rank $r\ll d$ from observations of its individual elements. Real-world matrix completion is often a huge-scale optimization problem, with $d$ so large that even the simplest full-dimension vector operations with $O(d)$ time complexity become prohibitively expensive. Stochastic gradient descent (SGD) is one of the few algorithms capable of solving matrix completion on a huge scale, and can also naturally handle streaming data over an evolving ground truth. Unfortunately, SGD experiences a dramatic slow-down when the underlying ground truth is ill-conditioned; it requires at least $O(\kappa\log(1/\epsilon))$ iterations to get $\epsilon$-close to ground truth matrix with condition number $\kappa$. In this paper, we propose a preconditioned version of SGD that preserves all the favorable practical qualities of SGD for huge-scale online optimization while also making it agnostic to $\kappa$. For a symmetric ground truth and the Root Mean Square Error (RMSE) loss, we prove that the preconditioned SGD converges to $\epsilon$-accuracy in $O(\log(1/\epsilon))$ iterations, with a rapid linear convergence rate as if the ground truth were perfectly conditioned with $\kappa=1$. In our numerical experiments, we observe a similar acceleration for ill-conditioned matrix completion under the 1-bit cross-entropy loss, as well as pairwise losses such as the Bayesian Personalized Ranking (BPR) loss.more » « less
-
null (Ed.)This survey describes probabilistic algorithms for linear algebraic computations, such as factorizing matrices and solving linear systems. It focuses on techniques that have a proven track record for real-world problems. The paper treats both the theoretical foundations of the subject and practical computational issues. Topics 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 factorizations, Nyström approximation of positive semidefinite matrices, single-view (‘streaming’) algorithms, full rank-revealing factorizations, solvers for linear systems, and approximation of kernel matrices that arise in machine learning and in scientific computing.more » « less
-
null (Ed.)A cumbersome operation in numerical analysis and linear algebra, optimization, machine learning and engineering algorithms; is inverting large full-rank matrices which appears in various processes and applications [1]. This has both numerical stability and complexity issues, as well as high expected time to compute. We address the latter issue, by proposing an algorithm which uses a black-box least squares optimization solver as a subroutine, to give an estimate of the inverse (and pseudoinverse) of real nonsingular matrices; by estimating its columns. This also gives it the flexibility to be performed in a distributed manner, thus the estimate can be obtained a lot faster, and can be made robust to stragglers. Furthermore, we assume a centralized network with no message passing between the computing nodes, and do not require a matrix factorization; e.g. LU, SVD or QR decomposition beforehand.more » « less
- Free Publicly Accessible Full Text
-
Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1002/cpe.7726
