Matrix Multiplication Over Word-Size Modular Rings Using Approximate Formulas

Boyer, Brice; Dumas, J.

doi:10.1145/2829947

Cited by 4 publications

(5 citation statements)

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“…Approximate matrix multiplication tensors were first introduced by Bini et ali in [2] in order to improve asymptotic bounds. From a practical point of view, these approximate algorithms could be used efficiently when the coefficients are in Z/𝑝Z (see [4]). Furthermore, from a theoretical point of view, these tensors allow to work with Euclidean closure of the Brent algebraic variety defined by Equation (19) and not the Zariski closure induced by dealing with exact tensors.…”

Section: New Approximate Algorithmsmentioning

confidence: 99%

The Tensor Rank of 5x5 Matrices Multiplication is Bounded by 98 andIts Border Rank by 89

Sedoglavic

Смирнов

2021

Proceedings of the 2021 International Symposium on Symbolic and Algebraic Computation

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We present a non-commutative algorithm for the product of 3 × 5 by 5 × 5 matrices using 58 multiplications. This algorithm allows to construct a non-commutative algorithm for multiplying 5 × 5 (resp. 10 × 10, 15 × 15) matrices using 98 (resp. 686, 2088) multiplications. Furthermore, we describe an approximate algorithm that requires 89 multiplications and computes this product with an arbitrary small error. CCS CONCEPTS• Computing methodologies → Exact arithmetic algorithms; Linear algebra algorithms.

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Section: New Approximate Algorithmsmentioning

confidence: 99%

The Tensor Rank of 5x5 Matrices Multiplication is Bounded by 98 andIts Border Rank by 89

Sedoglavic

Смирнов

2021

Proceedings of the 2021 International Symposium on Symbolic and Algebraic Computation

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“…For instance, for a product of dimension 12, with base case dimension b = 3, this gives; L A = L B = L C = (1,2,0,4,5,3,7,8,6,11,9,10) and K A = K B = K C = (0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11).…”

Section: Data Layout and Encryptionmentioning

confidence: 99%

“…For all i, j ∈ {1..n} 2 Parties We only give a sketch of the proof, since its very similar to the one for the MaskAndDecrypt protocol within the proof of Theorem 4.…”

Section: Protocol 6 Mp-mat-copymentioning

confidence: 99%

“…http://www.multipartycomputation.com/mpc-software5 The best value known to date, due to LeGall's[21], of approximately 2.3728639.However, only a few sub-cubic time algorithms are competitive in practice and used in software[9,2,19] (see also[20] and references therein), among which Strassen's algorithm and its variants stand out as a very effective one in practice.…”

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confidence: 99%

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Secure Multiparty Matrix Multiplication Based on Strassen-Winograd Algorithm

Dumas

Lafourcade

Fenner

et al. 2019

Advances in Information and Computer Security

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This paper presents a secure multiparty computation protocol for the Strassen-Winograd matrix multiplication algorithm. We focus on the setting in which any given player knows only one row (or one block of rows) of both input matrices and learns the corresponding row (or block of rows) of the resulting product matrix. Neither the player initial data, nor the intermediate values, even during the recurrence part of the algorithm, are ever revealed to other players. We use a combination of partial homomorphic encryption schemes and additive masking techniques together with a novel schedule for the location and encryption layout of all intermediate computations to preserve privacy. Compared to state of the art protocols, the asymptotic communication volume of our construction is reduced from O(n 3) to O(n 2.81). This improvement in terms of communication volume arises with matrices of dimension as small as n = 96 which is confirmed by experiments.

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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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Matrix Multiplication Over Word-Size Modular Rings Using Approximate Formulas

Cited by 4 publications

References 10 publications

The Tensor Rank of 5x5 Matrices Multiplication is Bounded by 98 andIts Border Rank by 89

The Tensor Rank of 5x5 Matrices Multiplication is Bounded by 98 andIts Border Rank by 89

Secure Multiparty Matrix Multiplication Based on Strassen-Winograd Algorithm

Fast Feasible and Unfeasible Matrix Multiplication

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