## Breaking the Coppersmith-Winograd barrier. Unpublished manuscript (2011)

Citations: | 10 - 0 self |

### BibTeX

@MISC{Williams11breakingthe,

author = {Virginia Vassilevska Williams},

title = {Breaking the Coppersmith-Winograd barrier. Unpublished manuscript},

year = {2011}

}

### OpenURL

### Abstract

We develop new tools for analyzing matrix multiplication constructions similar to the Coppersmith-Winograd construction, and obtain a new improved bound on ω < 2.3727. 1

### Citations

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Citation Context ... mathematics and computer science. Many other essential matrix operations can be efficiently reduced to it, such as Gaussian elimination, LUP decomposition, the determinant or the inverse of a matrix =-=[1]-=-. Matrix multiplication is also used as a subroutine in many computational problems that, on the face of it, have nothing to do with matrices. As a small sample illustrating the variety of application... |

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Citation Context ...rassen introduced his laser method which allowed for an entirely new attack on the matrix multiplication problem. He also decreased the bound to ω < 2.479. Three years later, Coppersmith and Winograd =-=[10]-=- combined Strassen’s technique with a novel form of analysis based on large sets avoiding arithmetic progressions and obtained the famous bound of ω < 2.376 which has remained unchanged for more than ... |

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Citation Context ...ces. As a small sample illustrating the variety of applications, there are faster algorithms relying on matrix multiplication for graph transitive closure (see e.g. [1]), context free grammar parsing =-=[20]-=-, and even learning juntas [13]. Until the late 1960s it was believed that computing the product C of two n × n matrices requires essentially a cubic number of operations, as the fastest algorithm kno... |

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Citation Context ...n the lemma. 1.1 A brief summary of the techniques used in bilinear matrix multiplication algorithms A full exposition of the techniques can be found in the book by Bürgisser, Clausen and Shokrollahi =-=[6]-=-. The lecture notes by Bläser [5] are also a nice read. Bilinear algorithms and trilinear forms. Matrix multiplication is an example of a trilinear form. n × n matrix multiplication, for instance, can... |

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Citation Context ...e than twenty years. In 2003, Cohn and Umans [8] introduced a new, group-theoretic framework for designing and analyzing matrix multiplication algorithms. In 2005, together with Kleinberg and Szegedy =-=[7]-=-, they obtained several novel matrix multiplication algorithms using the new framework, however they were not able to beat 2.376. Many researchers believe that the true value of ω is 2. In fact, both ... |

43 |
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(Show Context)
Citation Context ... In the same paper, combining his work with ideas by Pan, he also showed ω < 2.522. The following year, Romani [15] found that ω < 2.517. The first result to break 2.5 was by Coppersmith and Winograd =-=[9]-=- who obtained ω < 2.496. In 1986, Strassen introduced his laser method which allowed for an entirely new attack on the matrix multiplication problem. He also decreased the bound to ω < 2.479. Three ye... |

32 |
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Citation Context ...c progression is a sequence of three integers a ≤ b ≤ c so that b − a = c − b, or equivalently, a + c = 2b. An arithmetic progression is nontrivial if a < b < c. The following is a theorem by Behrend =-=[3]-=- improving on Salem and Spencer [16]. The subset A computed by the theorem is called a Salem-Spencer set. Theorem 1. There exists an absolute constant c such that for every N ≥ exp(c 2 ), one can cons... |

31 | A group-theoretic approach to fast matrix multiplication
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(Show Context)
Citation Context ...novel form of analysis based on large sets avoiding arithmetic progressions and obtained the famous bound of ω < 2.376 which has remained unchanged for more than twenty years. In 2003, Cohn and Umans =-=[8]-=- introduced a new, group-theoretic framework for designing and analyzing matrix multiplication algorithms. In 2005, together with Kleinberg and Szegedy [7], they obtained several novel matrix multipli... |

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Citation Context ...ting the variety of applications, there are faster algorithms relying on matrix multiplication for graph transitive closure (see e.g. [1]), context free grammar parsing [20], and even learning juntas =-=[13]-=-. Until the late 1960s it was believed that computing the product C of two n × n matrices requires essentially a cubic number of operations, as the fastest algorithm known was the naive algorithm whic... |

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Citation Context ...y reduced the matrix multiplication exponent ω over time. In 1978, Pan [14] showed ω < 2.796. The following year, Bini et al. [4] introduced the notion of border rank and obtained ω < 2.78. Schönhage =-=[17]-=- generalized this notion in 1981, proved his τ-theorem (also called the asymptotic sum inequality), and showed that ω < 2.548. In the same paper, combining his work with ideas by Pan, he also showed ω... |

25 |
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Citation Context ...e integers a ≤ b ≤ c so that b − a = c − b, or equivalently, a + c = 2b. An arithmetic progression is nontrivial if a < b < c. The following is a theorem by Behrend [3] improving on Salem and Spencer =-=[16]-=-. The subset A computed by the theorem is called a Salem-Spencer set. Theorem 1. There exists an absolute constant c such that for every N ≥ exp(c 2 ), one can construct in poly(N) time a subset A ⊂ [... |

24 |
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(Show Context)
Citation Context ...ime. This amazing discovery spawned a long line of research which gradually reduced the matrix multiplication exponent ω over time. In 1978, Pan [14] showed ω < 2.796. The following year, Bini et al. =-=[4]-=- introduced the notion of border rank and obtained ω < 2.78. Schönhage [17] generalized this notion in 1981, proved his τ-theorem (also called the asymptotic sum inequality), and showed that ω < 2.548... |

7 |
Some properties of disjoint sums of tensors related to matrix multiplication
- Romani
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(Show Context)
Citation Context ...ved his τ-theorem (also called the asymptotic sum inequality), and showed that ω < 2.548. In the same paper, combining his work with ideas by Pan, he also showed ω < 2.522. The following year, Romani =-=[15]-=- found that ω < 2.517. The first result to break 2.5 was by Coppersmith and Winograd [9] who obtained ω < 2.496. In 1986, Strassen introduced his laser method which allowed for an entirely new attack ... |

5 |
Strassen’s algorithm is not optimal
- Pan
- 1978
(Show Context)
Citation Context ...thm for matrix multiplication, running in O(n 2.808 ) time. This amazing discovery spawned a long line of research which gradually reduced the matrix multiplication exponent ω over time. In 1978, Pan =-=[14]-=- showed ω < 2.796. The following year, Bini et al. [4] introduced the notion of border rank and obtained ω < 2.78. Schönhage [17] generalized this notion in 1981, proved his τ-theorem (also called the... |

3 | On sunflowers and matrix multiplication - Alon, Shpilka, et al. |

3 |
Complexity of bilinear problems (lecture notes scribed by Fabian Bendun), http://www-cc.cs.uni-saarland.de/teaching/SS09/ComplexityofBilinearProblems/ script.pdf
- Bläser
- 2009
(Show Context)
Citation Context ...of the techniques used in bilinear matrix multiplication algorithms A full exposition of the techniques can be found in the book by Bürgisser, Clausen and Shokrollahi [6]. The lecture notes by Bläser =-=[5]-=- are also a nice read. Bilinear algorithms and trilinear forms. Matrix multiplication is an example of a trilinear form. n × n matrix multiplication, for instance, can be written as ∑ ∑ i,j∈[n] k∈n xi... |

1 |
Gaussian elimination is not optimal
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(Show Context)
Citation Context ...analyze larger tensor powers, and since the third tensor power does not give any improvement, the prospects looked bleak. Stothers’ work. We were recently made aware of the thesis work of A. Stothers =-=[18]-=- in which he claims an improvement to ω. Stothers argues that ω < 2.3737 by analyzing the 4th tensor power of the CoppersmithWinograd construction. Our approach can be seen as a vast generalization of... |