Barriers for rectangular matrix multiplication

Abstract We study the algorithmic problem of multiplying large matrices that are rectangular. We prove that the method that has been used to construct the fastest algorithms for rectangular matrix multiplication cannot give algorithms with complexity $$n^{p + 1}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>n</mml:mi> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> </mml:math> for $$n \times n$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>×</mml:mo> <mml:mi>n</mml:mi> </mml:mrow> </mml:math> by $$n \times n^p$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>×</mml:mo> <mml:msup> <mml:mi>n</mml:mi> <mml:mi>p</mml:mi> </mml:msup> </mml:mrow> </mml:math> matrix multiplication. In fact, we prove a precise numerical barrier for this method. Our barrier improves the previously known barriers, both in the numerical sense, as well as in its generality. In particular, we prove that any lower bound on the dual exponent of matrix multiplication $$\alpha$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>α</mml:mi> </mml:math> via the big Coppersmith-Winograd tensors cannot exceed $$0.6218$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mn>0.6218</mml:mn> </mml:mrow> </mml:math> .