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On Transformations of Algorithms to Multiply 2*2 Matrices.
Abstract
Representing these algorithms by noncommulative polynomials, we construct first the optimal algorithm of Winograd. Then, using an invariant transformation-in the algebraic sense-we transform this algorithm into the algorithm of Strassen. Follow remarks on this family of algorithms.
... The decomposition corresponding to Strassen's algorithm is then invariant under the cyclic permutation of matrices X, Y, Z. This symmetry is exploited in the proof of Chatelin [9], which uses properties of polynomials invariant under this symmetry. He also notices the importance of a matrix which is related to the S 3 symmetry discussed above. ...
The main purpose of this paper is pedagogical. Despite its importance, all proofs of the correctness of Strassen's famous 1969 algorithm to multiply two 2x2 matrices with only seven multiplications involve some basis-dependent calculations such as explicitly multiplying specific 2x2 matrices, expanding expressions to cancel terms with opposing signs, or expanding tensors over the standard basis. This makes the proof nontrivial to memorize and many presentations of the proof avoid showing all the details and leave a significant amount of verifications to the reader. In this note we give a short, self-contained, basis-independent proof of the existence of Strassen's algorithm that avoids these types of calculations. We achieve this by focusing on symmetries and algebraic properties. Our proof can be seen as a coordinate-free version of the construction of Clausen from 1988, combined with recent work on the geometry of Strassen's algorithm by Chiantini, Ikenmeyer, Landsberg, and Ottaviani from 2016.
... The decomposition corresponding to Strassen's algorithm is then invariant under the cyclic permutation of matrices X , Y , Z . This symmetry is exploited in the proof of Chatelin [9], which uses properties of polynomials invariant under this symmetry. He also notices the importance of a matrix which is related to the S 3 symmetry discussed above. ...
Despite its importance, all proofs of the correctness of Strassen's famous 1969 algorithm to multiply two 2x2 matrices with only seven multiplications involve some more or less tedious calculations such as explicitly multiplying specific 2x2 matrices, expanding expressions to cancel terms with opposing signs, or expanding tensors over the standard basis. This is why the proof is nontrivial to memorize and why many presentations of the proof avoid showing all the details and leave a significant amount of verifications to the reader. In this note we give a short, self-contained, easy to memorize, and elegant proof of the existence of Strassen's algorithm that avoids these types of calculations. We achieve this by focusing on symmetries and algebraic properties. Our proof combines the classical theory of M-pairs, which was initiated by B\"uchi and Clausen in 1985, with recent work on the geometry of Strassen's algorithm by Chiantini, Ikenmeyer, Landsberg, and Ottaviani from 2016.
... We keep the parameter λ useless in our presentation as a tribute to the construction made in [7] that gives an elegant and elementary (i.e. based on matrix eigenvalues) construction of Winograd variant of Strassen matrix multiplication algorithm. This variant is remarkable in its own as shown in [3] because it is optimal w.r.t. ...
In~1969, V. Strassen improves theclassical~2x2 matrix multiplication algorithm. Thecurrent upper bound for~3x3 matrix multiplication wasreached by J.B.\ Laderman in 1976.This note presents a geometric relationship between Strassen andLaderman algorithms.
We present a non-commutative algorithm for the multiplication of a block-matrix by its transpose over C or any finite field using 5 recursive products. We use geometric considerations on the space of bilinear forms describing 2x2 matrix products to obtain this algorithm and we show how to reduce the number of involved additions. The resulting algorithm for arbitrary dimensions is a reduction of multiplication of a matrix by its transpose to general matrix product, improving by a constant factor previously known reductions. Finally we propose space and time efficient schedules that enable us to provide fast practical implementations for higher-dimensional matrix products.
In 1969, Strassen shocked the world by showing that two n x n matrices could be multiplied in time asymptotically less than . While the recursive construction in his algorithm is very clear, the key gain was made by showing that 2 x 2 matrix multiplication could be performed with only 7 multiplications instead of 8. The latter construction was arrived at by a process of elimination and appears to come out of thin air. Here, we give the simplest and most transparent proof of Strassen's algorithm that we are aware of, using only a simple unitary 2-design and a few easy lines of calculation. Moreover, using basic facts from the representation theory of finite groups, we use 2-designs coming from group orbits to generalize our construction to all n (although the resulting algorithms aren't optimal for n at least 3).
Nonstandard transformations over matrices allow the derivation of different, optimal algorithms in digital signal processing.
t. Below we will give an algorithm which computes the coefficients of the product of two square matrices A and B of order n from the coefficients of A and B with tess than 4.7-n l°g7 arithmetical operations (all logarithms in this paper are for base 2, thus tog 7 ~ 2.8; the usual method requires approximately 2n 3 arithmetical operations). The algorithm induces algorithms for inverting a matrix of order n, solving a system of n linear equations in n unknowns, computing a determinant of order n etc. all requiring less than const n l°g 7 arithmetical operations. This fact should be compared with the result of KLYUYEV and KOKOVKINSHCHERBAK [1 ] that Gaussian elimination for solving a system of linearequations is optimal if one restricts oneself to operations upon rows and columns as a whole. We also note that WlNOGRAD [21 modifies the usual algorithms for matrix multiplication and inversion and for solving systems of linear equations, trading roughly half of the multiplications for additions and subtractions. It is a pleasure to thank D. BRILLINGER for inspiring discussions about the present subject and ST. COOK and B. PARLETT for encouraging me to write this paper. 2. We define algorithms e~, ~ which multiply matrices of order m2 ~, by induction on k: ~,0 is the usual algorithm, for matrix multiplication (requiring m a multiplications and m 2 (m- t) additions), e~,k already being known, define ~, ~ +t as follows: If A, B are matrices of order m 2 k ~ to be multiplied, write (All A~2 t (B~I B12~
This paper contains the general frame of a theory of varieties of algorithms for the computation of bilinear mappings. In particular, we shall study linear mappings which operate on algo0rithm varieties. 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. This group is called the isotropy group of Φ. For some important classes of bilinear mappings these groups will be determined. Applications of these results will appear in parts II and III of this work.
In this paper we will show that Strassen's algorithm for the computation of the product of 2 × 2-matrices is essentially unique. As an application of this result we will answer the question to what extent elements of the trivial algorithm for 2 × 2-matrix multiplication can be used in an optimal one.
Sprcifications et manipulations des pro-grammes: cas d'un ensemble de formes bilinraires Juin 1979. I refer to the original version of Strassen's algorithm in
- [ 1}
- P Chatelin
[1} P. Chatelin, Sprcifications et manipulations des pro-grammes: cas d'un ensemble de formes bilinraires, Thrse, Universit6 de Grenoble, Juin 1979. I refer to the original version of Strassen's algorithm in [2]:
Spécifications et manipulations des programmes: cas d'un ensemble de formes bilinéaires
- Chatelin