New lower bounds for the border rank of matrix multiplication

Details New lower bounds for the border rank of matrix multiplication New lower bounds for the border rank of matrix multiplication

2011-12-27

arxiv.org/abs/1112.6007v2

Computer Science

Computational Complexity

9 pages. Version 1 contained an error in the proof of its main theorem and in the course of fixing it we proved a stronger sta

Scientific paper

The border rank of the matrix multiplication operator for n by n matrices is a standard measure of its complexity. Using techniques from algebraic geometry and representation theory, we show the border rank is at least 2n^2-n. Our bounds are better than the previous lower bound (due to Lickteig in 1985) of 3/2 n^2+ n/2 -1 for all n>2. The bounds are obtained by finding new equations that bilinear maps of small border rank must satisfy, i.e., new equations for secant varieties of triple Segre products, that matrix multiplication fails to satisfy.

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