133,628 research outputs found
Plethysm and fast matrix multiplication
Motivated by the symmetric version of matrix multiplication we study the
plethysm of the adjoint representation
of the Lie group . In particular, we describe the decomposition of this
representation into irreducible components for , and find highest weight
vectors for all irreducible components. Relations to fast matrix
multiplication, in particular the Coppersmith-Winograd tensor are presented.Comment: 5 page
Fast matrix multiplication techniques based on the Adleman-Lipton model
On distributed memory electronic computers, the implementation and
association of fast parallel matrix multiplication algorithms has yielded
astounding results and insights. In this discourse, we use the tools of
molecular biology to demonstrate the theoretical encoding of Strassen's fast
matrix multiplication algorithm with DNA based on an -moduli set in the
residue number system, thereby demonstrating the viability of computational
mathematics with DNA. As a result, a general scalable implementation of this
model in the DNA computing paradigm is presented and can be generalized to the
application of \emph{all} fast matrix multiplication algorithms on a DNA
computer. We also discuss the practical capabilities and issues of this
scalable implementation. Fast methods of matrix computations with DNA are
important because they also allow for the efficient implementation of other
algorithms (i.e. inversion, computing determinants, and graph theory) with DNA.Comment: To appear in the International Journal of Computer Engineering
Research. Minor changes made to make the preprint as similar as possible to
the published versio
Powers of Tensors and Fast Matrix Multiplication
This paper presents a method to analyze the powers of a given trilinear form
(a special kind of algebraic constructions also called a tensor) and obtain
upper bounds on the asymptotic complexity of matrix multiplication. Compared
with existing approaches, this method is based on convex optimization, and thus
has polynomial-time complexity. As an application, we use this method to study
powers of the construction given by Coppersmith and Winograd [Journal of
Symbolic Computation, 1990] and obtain the upper bound on
the exponent of square matrix multiplication, which slightly improves the best
known upper bound.Comment: 28 page
Group-theoretic algorithms for matrix multiplication
We further develop the group-theoretic approach to fast matrix multiplication
introduced by Cohn and Umans, and for the first time use it to derive
algorithms asymptotically faster than the standard algorithm. We describe
several families of wreath product groups that achieve matrix multiplication
exponent less than 3, the asymptotically fastest of which achieves exponent
2.41. We present two conjectures regarding specific improvements, one
combinatorial and the other algebraic. Either one would imply that the exponent
of matrix multiplication is 2.Comment: 10 page
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