Adaptive Winograd's matrix multiplications

D'Alberto, Paolo; Nicolau, Alexandru

doi:10.1145/1486525.1486528

Cited by 20 publications

(7 citation statements)

References 43 publications

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“…The implementation of these old algorithms has been extensively worked on by many authors and makes up a valuable part of the present day MM software (cf. [14], [81], [63], [68], [69], [34], [56], [8], [18], [33], the references therein, and in [78,Chapter 1]). This work, intensified lately, is still mostly devoted to the implementation of very old algorithms, ignoring, for example, the advanced implementations of fast MM in [90] and [91] (see Section 17.1) and the significant improvement in [45] and [98] We hope that our survey will motivate advancing the State of the Art both in the design of fast algorithms for feasible MM and in their efficient implementation.…”

Section: Numerical Implementation Of Fast MMmentioning

confidence: 99%

Fast Feasible and Unfeasible Matrix Multiplication

Pan

2018

Preprint

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Matrix-by-matrix multiplication (hereafter referred to as MM) is a fundamental operation omnipresent in modern computations in Numerical and Symbolic Linear Algebra. Its acceleration makes major impact on various fields of Modern Computation and has been a highly recognized research subject for about five decades. The researchers introduced amazing novel techniques, found new insights into MM and numerous related computational problems, and devised advanced algorithms that performed n × n MM by using less than O(n 2.38 ) scalar arithmetic operations versus 2n 3 − n 2 of the straightforward MM, that is, more than half-way to the information lower bound n 2 . The record upper bound 3 of 1968 on the exponent of the complexity MM decreased below 2.38 by 1987 and has been extended to various celebrated problems in many areas of computing and became most extensively cited constant of the Theory of Computing. The progress in decreasing the record exponent, however, has virtually stalled since 1987, while many scientists are still anxious to know its sharp bound, so far restricted to the range from 2 to about 2.3728639. Narrowing this range remains a celebrated challenge.Acceleration of MM in the Practice of Computing is a distinct challenge, much less popular, but also highly important. Since 1980 the progress towards meeting the two challenges has followed two distinct paths because of the curse of recursion -all the known algorithms supporting the exponents below 2.38 or even below 2.7733 involve long sequences of nested recursive steps, which blow up the size of an input matrix. As a result all these algorithms improve straightforward MM only for unfeasible MM of immense size, greatly exceeding the sizes of interest nowadays and in any foreseeable future.It is plausible and surely highly desirable that someone could eventually decrease the record MM exponent towards its sharp bound 2 even without ignoring the curse of recursion, but currently there are two distinct challenges of the acceleration of feasible and unfeasible MM.In particular various known algorithms supporting the exponents in the range between 2.77 and 2.81 are quite efficient for feasible MM and have been implemented. Some of them make up a valuable part of modern software for numerical and symbolic matrix computations, extensively worked on in the last decade. Still, that work has mostly relied on the MM algorithms proposed more than four decades ago, while more efficient algorithms are well known, some of them appeared in 2017.In our review we first survey the mainstream study of the acceleration of MM of unbounded sizes, cover the progress in decreasing the exponents of MM, comment on its impact on the theory and practice of computing, and recall various fundamental concepts and techniques supporting fast MM and naturally introduced in that study by 1980. Then we demonstrate how the curse of recursion naturally entered the game of decreasing the record exponents. Finally we cover the State of the Art of efficient feasible MM, including some most ef...

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Section: Numerical Implementation Of Fast MMmentioning

confidence: 99%

Fast Feasible and Unfeasible Matrix Multiplication

Pan

2018

Preprint

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“…In Section 1.3 we commented on numerical stability issues for fast feasible MM, and we refer the reader to [12], [73], [52], [61], [62], [28], [45], [71, Chapter 1], [13], [8], [11], and the bibliography therein for their previous and current numerical implementation. In the next section we discuss symbolic application of the latter algorithm (WRB-MM) in some detail.…”

Section: Summary Of the Study Of The MM Exponents After 1978mentioning

confidence: 99%

Fast Matrix Multiplication and Symbolic Computation

Dumas,

Pan

2016

Preprint

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The complexity of matrix multiplication (hereafter MM) has been intensively studied since 1969, when Strassen surprisingly decreased the exponent 3 in the cubic cost of the straightforward classical MM to log 2 (7) ≈ 2.8074. Applications to some fundamental problems of Linear Algebra and Computer Science have been immediately recognized, but the researchers in Computer Algebra keep discovering more and more applications even today, with no sign of slowdown. We survey the unfinished history of decreasing the exponent towards its information lower bound 2, recall some important techniques discovered in this process and linked to other fields of computing, reveal sample surprising applications to fast computation of the inner products of two vectors and summation of integers, and discuss the curse of recursion, which separates the progress in fast MM into its most acclaimed and purely theoretical part and into valuable acceleration of MM of feasible sizes. Then, in the second part of our paper, we cover fast MM in realistic symbolic computations and discuss applications and implementation of fast exact matrix multiplication. We first review how most of exact linear algebra can be reduced to matrix multiplication over small finite fields. Then we highlight the differences in the design of approximate and exact implementations of fast MM, taking into account nowadays processor and memory hierarchies. In the concluding section we comment on current perspectives of the study of fast MM.

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“…When the matrix's dimension is larger than the recursion point, the Strassen/Winograd's MM is more appealing. D'alberto et al provides the parallel implementation of fast MM algorithms in Strassen's, Winograd's, and 3M to further improve the speed of matrix multiplication. These general‐purpose fast MM schemes can be seen as the bases of our work in which we reduce the dimensions of matrices and render them to these MM schemes to accomplish matrix disguising.…”

Section: Related Workmentioning

confidence: 99%

An efficient and tunable matrix-disguising method toward privacy-preserving computation

Wang

2015

Security Comm. Networks

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A matrix is a basic mathematical object that is widely used in various computations. When outsourcing expensive computations to untrusted parties, the involved matrix must be disguised before it's sent out in order to protect the privacy information in it. Some research works on secure computation had presented schemes for protecting the privacy in matrices. However, none of these schemes is defined deliberately for disguising a matrix and thus is neither highly efficient nor flexible. We propose a matrix‐disguising method named FMD (fast matrix disguising) that has high time and space efficiency and can tune the trade‐off between disguising speed and protecting strength with a parameter. FMD disguises a matrix by multiplying it with a semi‐random non‐singular matrix which is compose of many bar‐shaped sub‐matrices. Each of these sub‐matrices contains a row/column of random elements with almost the same values. This special matrix structure allows FMD to disguise the original matrix with time complexity proportional to the size of the original matrix. While by adjusting the bar size of the sub‐matrices, FMD can smoothly tune between high‐disguising speed and high‐privacy protection strength. The mathematical analysis and experimental results show that FMD is more efficient than the existing schemes and is especially suitable for resource‐limited clients in privacy‐preserving computation outsourcing scenarios. Copyright © 2015 John Wiley & Sons, Ltd.

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Adaptive Winograd's matrix multiplications

Cited by 20 publications

References 43 publications

Fast Feasible and Unfeasible Matrix Multiplication

Fast Feasible and Unfeasible Matrix Multiplication

Fast Matrix Multiplication and Symbolic Computation

An efficient and tunable matrix-disguising method toward privacy-preserving computation

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