On varieties of optimal algorithms for the computation of bilinear mappings II. optimal algorithms for 2 × 2-matrix multiplication
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TL;DR: It is shown that Strassen's algorithm for the computation of the product of 2 × 2-matrices is essentially unique, and the question to what extent elements of the trivial algorithm for 2 ×2-matrix multiplication can be used in an optimal one is answered.
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About: This article is published in Theoretical Computer Science. The article was published on 01 Jan 1978. and is currently open access. The article focuses on the topics: Strassen algorithm & Matrix multiplication.
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Citations
Discovering faster matrix multiplication algorithms with reinforcement learning
Alhussein Fawzi,Matej Balog,Aja Huang,Thomas Hubert,Bernardino Romera-Paredes,Mohammadamin Barekatain,Alexander Novikov,Francisco J. R. Ruiz,Julian Schrittwieser,Grzegorz Swirszcz,David Silver,Demis Hassabis,Pushmeet Kohli +12 more
TL;DR: In this paper , a deep reinforcement learning approach based on AlphaZero is used to discover efficient and provably correct algorithms for the multiplication of arbitrary matrices, where the objective is finding tensor decompositions within a finite factor space.
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Andrew James Stothers
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TL;DR: In this article, it was shown that ω ≤ log 2 7 < 2.8074, which is better than the value of 3 we had previously, and showed how cubing and raising to the fourth power of Coppersmith and Winograd's complicated algorithm can improve the precision of matrix multiplication.
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Relative bilinear complexity and matrix multiplication.
TL;DR: The significance of this notion lies, above all, in the key role of matrix multiplication for numerical linear algebra, where the following problems all have "exponent'": Matrix inversion, LK-decomposition, evaluation of the determinant or of all coefficients of the characteristic polynomial and for k = C also Qß- decomposition and unitary transformation to Hessenberg form.
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Fast Matrix Multiplication
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Geometry and Complexity Theory
Joseph M. Landsberg
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TL;DR: This introduction to algebraic complexity theory for graduate students and researchers in computer science and mathematics features concrete examples that demonstrate the application of geometric techniques to real world problems.
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References
Gaussian elimination is not optimal
TL;DR: In this paper, Cook et al. gave an algorithm which computes the coefficients of the product of two square matrices A and B of order n with less than 4. 7 n l°g 7 arithmetical operations (all logarithms in this paper are for base 2).
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On multiplication of 2 × 2 matrices
TL;DR: In this paper, the minimum number of multiplications required to multiply two 2 X 2 matrices is seven and three, respectively, and the number of complex numbers required is three.
265
The Complexity of the Quaternion Product
Thomas D. Howell,Jean Lafon +1 more
- 01 Jun 1975
TL;DR: In this article, it was shown that at least seven multiplications are necessary and sufficient for computing the product of two quaternions over an arbitrary ring if the ring is commutative.
On varieties of optimal algorithms for the computation of bilinear mappings I. the isotropy group of a bilinear mapping
TL;DR: It will be shown that every bilinear mapping Φ defines in a natural way a group of automorphisms operating on the variety of optimal algorithms for Φ, called the isotropy group of Φ.