New Lower Bounds for the Rank of Matrix Multiplication
نویسنده
چکیده
The rank of the matrix multiplication operator for n×n matrices is one of the most studied quantities in algebraic complexity theory. I prove that the rank is at least 3n2−o(n2). More precisely, for any integer p ≤ n− 1 the rank is at least (3− 1 p+1 )n2 − (1 + 2p 2p p−1 ) )n. The previous lower bound, due to Bläser, was 5 2 n2−3n (the case p = 1). The new bounds improve Bläser’s bound for all n > 84. I also prove lower bounds for rectangular matrices that are significantly better than the previous bound.
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عنوان ژورنال:
- SIAM J. Comput.
دوره 43 شماره
صفحات -
تاریخ انتشار 2014