Abstract
In 1979 considerable progress was made in estimating the complexity of matrix multiplication. Here the new techniques and recent results are presented, based upon the notion of approximate rank and the observation that certain patterns of partial matrix multiplication (some of the entries of the matrices may be zero) can efficiently be utilized to perform multiplication of large total matrices. By combining Pan’s trilinear technique with a strong version of our 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 $O(N^\beta )$, where $\beta $ is a bit smaller than $3\ln 52/\ln 110 \approx 2.522$.
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