Group-theoretic Algorithms for Matrix Multiplication

Cohn, Henry; Kleinberg, Robert; Szegedy, Balázs; Umans, Christopher

doi:10.1109/sfcs.2005.39

Cited by 137 publications

(251 citation statements)

References 8 publications

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“…In this section, we perform forward error analysis for three classes of recursive matrix multiplication algorithms, starting with the Strassen-like algorithms based on stationary partitioning, then generalizing to algorithms with non-stationary partitioning, and finally to the algorithms of the kind developed in [9] and [7]. The error analysis in Sections 3.1 and 3.2 is done with respect to the entry-wise maximum norm on A, B, C = AB, while the analysis in Section 3.3 is for an arbitrary matrix norm satisfying an extra monotonicity assumption.…”

Section: Model Of Arithmetic and Algorithmsmentioning

confidence: 99%

“…The complexity of the fastest known method to date due to D. Coppersmith and S. Winograd [10] is about O(n 2.38 ). A new approach based on group-theoretic methods was recently developed in [9] and [7], along with several ideas that can potentially reduce the bound on ω to ω = 2 (obviously, ω cannot fall below 2, since O(n 2 ) operations are required just to read off the entries of the resulting matrix).…”

Section: Introductionmentioning

confidence: 99%

“…It is the purpose of this work to analyze recursive fast matrix multiplication algorithms generalizing Strassen's algorithm, as well as the new class of algorithms described in [9] and [7], from the stability point of view. The rounding error analysis of Strassen's method was initiated by Brent ([4, 13], [14, chap.…”

Section: Introductionmentioning

confidence: 99%

“…We then generalize our analysis to algorithms where the number of blocks depends on the level of recursion, and finally to algorithms involving additional preprocessing before and postprocessing after partitioning into blocks. In Section 4 we use this approach to analyze the group-theoretic algorithms from [9] and [7]. In particular, in Section 4.4 we perform detailed stability analysis for three specific classes of algorithms introduced in [9] and [7].…”

Section: Introductionmentioning

confidence: 99%

“…In Section 4 we use this approach to analyze the group-theoretic algorithms from [9] and [7]. In particular, in Section 4.4 we perform detailed stability analysis for three specific classes of algorithms introduced in [9] and [7]. The paper ends with a brief discussion of other fast linear algebra algorithms in Section 5.…”

Section: Introductionmentioning

confidence: 99%

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Fast matrix multiplication is stable

et al. 2007

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We perform forward error analysis for a large class of recursive matrix multiplication algorithms in the spirit of [D. Bini and G. Lotti, Stability of fast algorithms for matrix multiplication, Numer. Math. 36 (1980), 63-72]. As a consequence of our analysis, we show that the exponent of matrix multiplication (the optimal running time) can be achieved by numerically stable algorithms. We also show that new group-theoretic algorithms proposed in [H. Cohn, and C. Umans, A group-theoretic approach to fast matrix multiplication, FOCS 2003, 438-449] and [H. Cohn, R. Kleinberg, B. Szegedy and C. Umans, Group-theoretic algorithms for matrix multiplication, FOCS 2005, 379-388] are all included in the class of algorithms to which our analysis applies, and are therefore numerically stable. We perform detailed error analysis for three specific fast group-theoretic algorithms.

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Section: Model Of Arithmetic and Algorithmsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Fast matrix multiplication is stable

et al. 2007

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An approach of performance comparisons with OpenMP and CUDA parallel programming on multicore systems

Chang

Yang

et al. 2016

Concurrency and Computation

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SUMMARYIn the past, the tenacious semiconductor problems of operating temperature and power consumption limited the performance growth for single-core microprocessors. Microprocessor vendors hence adopt the multicore chip organizations with parallel processing because the new technology promises faster and lower power needed. In a short time, this trend floods first the development of CPU, then also the other peripherals like GPU. Modern GPUs are very efficient in manipulating computer graphics, and their highly parallel structure makes them even more effective than general-purpose CPUs for a range of graphical complex algorithms. However, technology of multicore processor brought revolution and unavoidable collision to the programming personnel. Multicore processor has high performance; however, parallel processing brings not only the opportunity but also a challenge. The issue of efficiency and the way how programmer or compiler parallelizes the software explicitly are the keys that enhance the performance on multicore chip. In this paper, we propose a parallel programming approach using hybrid CUDA, OpenMP, and MPI programming. There would be two verificational experiments presented in the paper. In the first, we would verify the availability and correctness of the auto-parallel tools, and discuss the performance issues on CPU, GPU, and embedded system. In the second, we would verify how the hybrid programming could surely improve performance.

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Necklaces, Convolutions, and X + Y

Bremner

Chan

Demaine

et al. 2006

Lecture Notes in Computer Science

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In memory of our colleague Mihai Pǎtraşcu. AbstractWe give subquadratic algorithms that, given two necklaces each with n beads at arbitrary positions, compute the optimal rotation of the necklaces to best align the beads. Here alignment is measured according to the ℓ p norm of the vector of distances between pairs of beads from opposite necklaces in the best perfect matching. We show surprisingly different results for p = 1, p even, and p = ∞. For p even, we reduce the problem to standard convolution, while for p = ∞ and p = 1, we reduce the problem to (min, +) convolution and (median, +) convolution. Then we solve the latter two convolution problems in subquadratic time, which are interesting results in their own right. These results shed some light on the classic sorting X + Y problem, because the convolutions can be viewed as computing order statistics on the antidiagonals of the X + Y matrix. All of our algorithms run in o(n 2 ) time, whereas the obvious algorithms for these problems run in Θ(n 2 ) time.

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Group-theoretic Algorithms for Matrix Multiplication

Cited by 137 publications

References 8 publications

Fast matrix multiplication is stable

Fast matrix multiplication is stable

An approach of performance comparisons with OpenMP and CUDA parallel programming on multicore systems

Necklaces, Convolutions, and X + Y

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