Theory of Computing
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Title : New Lower Bounds for the Border Rank of Matrix Multiplication
Authors : Joseph M. Landsberg and Giorgio Ottaviani
Volume : 11
Number : 11
Pages : 285-298
URL : https://theoryofcomputing.org/articles/v011a011
Abstract
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The border rank of the matrix multiplication operator for
$n\times n$ 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 $3n^2/2 + n/2 - 1$ for
all $n\geq 3$. 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.