Kronecker scaling of tensors with applications to arithmetic circuits and algorithms

Abstract

We show that sufficiently low tensor rank for the balanced tripartitioning tensor $P_d(x,y,z)=\sum_{A,B,C\in\binom{[3d]}{d}:A\cup B\cup C=[3d]}x_Ay_Bz_C$ for a large enough constant $d$ implies uniform arithmetic circuits for the matrix permanent that are exponentially smaller than circuits obtainable from Ryser's formula. We show that the same low-rank assumption implies exponential time improvements over the state of the art for a wide variety of other related counting and decision problems. As our main methodological contribution, we show that the tensors $P_n$ have a desirable Kronecker scaling property: They can be decomposed efficiently into a small sum of restrictions of Kronecker powers of $P_d$ for constant $d$. We prove this with a new technique relying on Steinitz's lemma, which we hence call Steinitz balancing. As a consequence of our methods, we show that the mentioned low rank assumption (and hence the improved algorithms) is implied by Strassen's asymptotic rank conjecture [Progr. Math. 120 (1994)], a bold conjecture that has recently seen intriguing progress.