Multiplying matrices faster than coppersmith-winograd
@inproceedings{Williams2012MultiplyingMF, title={Multiplying matrices faster than coppersmith-winograd}, author={Virginia Vassilevska Williams}, booktitle={STOC '12}, year={2012} }
We develop an automated approach for designing matrix multiplication algorithms based on constructions similar to the Coppersmith-Winograd construction. Using this approach we obtain a new improved bound on the matrix multiplication exponent ω<2.3727.
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References
SHOWING 1-10 OF 29 REFERENCES
Matrix multiplication via arithmetic progressions
- MathematicsSTOC
- 1987
A new method for accelerating matrix multiplication asymptotically is presented, by using a basic trilinear form which is not a matrix product, and making novel use of the Salem-Spencer Theorem.
Group-theoretic algorithms for matrix multiplication
- Computer Science, Mathematics46th Annual IEEE Symposium on Foundations of Computer Science (FOCS'05)
- 2005
The group-theoretic approach to fast matrix multiplication introduced by Cohn and Umans is developed, and for the first time it is used to derive algorithms asymptotically faster than the standard algorithm.
On the Asymptotic Complexity of Matrix Multiplication
- Computer Science, MathematicsSIAM J. Comput.
- 1982
A consequence of these results is that $\omega $, the exponent for matrix multiplication, is a limit point, that is, it cannot be realized by any single algorithm.
Strassen's algorithm is not optimal trilinear technique of aggregating, uniting and canceling for constructing fast algorithms for matrix operations
- Computer Science19th Annual Symposium on Foundations of Computer Science (sfcs 1978)
- 1978
A new technique of trilinear operations of aggregating, uniting and canceling is introduced and applied to constructing fast linear non-commutative algorithms for matrix multiplication. The result is…
General Context-Free Recognition in Less than Cubic Time
- Computer ScienceJ. Comput. Syst. Sci.
- 1975
Partial and Total Matrix Multiplication
- Mathematics, Computer ScienceSIAM J. Comput.
- 1981
By combining Pan’s trilinear technique with a strong version of the compression theorem for the case of several disjoint matrix multiplications it is shown that multiplication of N \times N matrices (over arbitrary fields) is possible in time.
A group-theoretic approach to fast matrix multiplication
- Mathematics44th Annual IEEE Symposium on Foundations of Computer Science, 2003. Proceedings.
- 2003
A new, group-theoretic approach to bounding the exponent of matrix multiplication is developed, including a proof that certain families of groups of order n/sup 2+o(1)/ support n /spl times/ n matrix multiplication.
Relative bilinear complexity and matrix multiplication.
- Computer Science
- 1987
The significance of this notion lies, above all, in the key role of matrix multiplication for numerical linear algebra. Thus the following problems all have "exponent' : Matrix inversion,…
Some Properties of Disjoint Sums of Tensors Related to Matrix Multiplication
- MathematicsSIAM J. Comput.
- 1982
Let t be a disjoint sum of tensors associated to matrix multiplication. The rank of the tensorial powers of t is bounded by an expression involving the elements of t and an exponent for matrix…