Abstract
$ \newcommand\nnn{\mathbf{n}} $ The border rank of the matrix multiplication operator for $\nnn\times \nnn$ matrices is a standard measure of its complexity. Using techniques from algebraic geometry and representation theory, we show the border rank is at least $2\nnn^2-\nnn$. Our bounds are better than the previous lower bound (due to Lickteig in 1985) of $3\nnn^2/2+ {\nnn}/{2}-1$ for all $\nnn\geq 3$. 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.
Highlights
Introduction and statement of resultsFinding lower bounds in complexity theory is considered difficult
The complexity of matrix multiplication is roughly equivalent to the complexity of many standard operations in linear algebra, such as taking the determinant or inverse of a matrix
Our method is similar in nature to the method used by Strassen to get his lower bounds—we find explicit polynomials that tensors of low border rank must satisfy, and show that matrix multiplication fails to satisfy them
Summary
Finding lower bounds in complexity theory is considered difficult. For example, chapter 14 of [1] is titled “Circuit lower bounds: Complexity theory’s Waterloo.” The complexity of matrix multiplication is roughly equivalent to the complexity of many standard operations in linear algebra, such as taking the determinant or inverse of a matrix. Our method is similar in nature to the method used by Strassen to get his lower bounds—we find explicit polynomials that tensors of low border rank must satisfy, and show that matrix multiplication fails to satisfy them. Strassen found his equations via linear algebra—taking the commutator of certain matrices. An appendix (§A) with basic facts from representation theory that we use is included for readers not familiar with the subject
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