矩阵乘法指数界的群论方法
Group-theoretic Methods for Bounding the Exponent of Matrix Multiplication
摘要：
的矩阵乘法的（渐近）复杂性（在复数域上）是由一个实参数w > 0来度量的，称为矩阵乘法的指数（在复数域上），其被定义为最小实数w > 0，使得对于任意精度> 0，两个n乘n的复矩阵可以使用一种算法相乘，该算法使用O（n^（w+ε））个非除法算术运算。根据两个矩阵相乘的标准算法，指数w的平凡下界和上界分别为2和3。W. Strassen在1969年获得了第一个重要的结果，w < 2.81使用他的结果，2乘2矩阵乘法可以使用7乘法，而不是8，在标准算法。1984年，V. Pan使用Strassen方法的一个变体将其改进为2.67。已经证明了w = 2，但最著名的结果是w < 2.38，这是由于D. Coppersmith和S.威诺格拉德在所有这些方法中，对w的估计取决于其算法中主要运行步骤的数量。在2003年和2005年的一系列论文中，H. Cohn和C.乌曼斯提出了一个完全不同的方法使用相当初级的方法涉及有限群，群代数和他们的代表。作者描述和证明了他们的主要结果，并提出了利用他们的方法得到指数改进估计的可能途径。
英文摘要：
The (asymptotic) complexity of matrix multiplication (over the complex field) is measured by a real parameter w > 0, called the exponent of matrix multiplication (over the complex field), which is defined to be the smallest real number w > 0 such that for an arbitrary degree of precision > 0, two n by n complex matrices can be multiplied using an algorithm using O(n^(w+ε)) number of non-division arithmetical operations. By the standard algorithm for multiplying two matrices, the trivial lower and upper bounds for the exponent w are 2 and 3 respectively. W. Strassen in 1969 obtained the first important result that w < 2.81 using his result that 2 by 2 matrix multiplication could be performed using 7 multiplications, not 8, as in the standard algorithm. In 1984, V. Pan improved this to 2.67, using a variant of Strassen's approach. It has been conjectured that w = 2, but the best known result is that w < 2.38, due to D. Coppersmith and S. Winograd. In all these approaches, estimates for w depend on the number of main running steps in their algorithms. In a recent series of papers in 2003 and 2005, H. Cohn and C. Umans put forward an entirely different approach using fairly elementary methods involving finite groups, group algebras and their representations. The author describes and proves their main results, and suggests possible ways of getting improved estimates for the exponent using their methods.
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