Applications of Monotone Rank to Complexity Theory

Abstract

Raz's recent result Raz2010 has rekindled people's interest in the study of tensor rank, the generalization of matrix rank to high dimensions, by showing its connections to arithmetic formulas. In this paper, we follow Raz's work and show that monotone rank, the monotone variant of tensor rank and matrix rank, has applications in algebraic complexity, quantum computing and communication complexity. This paper differs from Raz's paper in that it leverages existing results to show unconditional bounds while Raz's result relies on some assumptions. We show a super-exponential separation between monotone and non-monotone computation in the non-commutative model, and thus provide a strong solution to Nisan's question Nis1991 in algebraic complexity. More specifically, we exhibit that there exists a homogeneous algebraic function $f$ of degree $d$ ($d$ even) on $n$ variables with the monotone algebraic branching program (ABP) complexity $Ω(d^2\log n)$