On Finer Separations between Subclasses of Read-once Oblivious ABPs

TL;DR

This paper investigates subclasses of Read-once Oblivious Algebraic Branching Programs (ROABPs) based on the algebraic properties of their coefficient matrices, revealing connections to depth-three powering circuits and polynomial ranks.

Contribution

It establishes a surprising link between commutative and diagonal ROABPs and depth-three powering circuits, highlighting the role of algebraic structure in computational complexity.

Findings

Diagonal ROABPs can simulate commutative ROABPs under certain conditions.

A super-polynomial separation exists if commutative ROABPs are not simulatable by diagonal ROABPs.

The algebraic structure of coefficient matrices is crucial for understanding ROABP subclasses.

Abstract

Read-once Oblivious Algebraic Branching Programs (ROABPs) compute polynomials as products of univariate polynomials that have matrices as coefficients. In an attempt to understand the landscape of algebraic complexity classes surrounding ROABPs, we study classes of ROABPs based on the algebraic structure of these coefficient matrices. We study connections between polynomials computed by these structured variants of ROABPs and other well-known classes of polynomials (such as depth-three powering circuits, tensor-rank and Waring rank of polynomials). Our main result concerns commutative ROABPs, where all coefficient matrices commute with each other, and diagonal ROABPs, where all the coefficient matrices are just diagonal matrices. In particular, we show a somewhat surprising connection between these models and the model of depth-three powering circuits that is related to the Waring rank of polynomials. We show that if the dimension of partial derivatives captures Waring rank up to polynomial factors, then the model of diagonal ROABPs efficiently simulates the seemingly more expressive model of commutative ROABPs. Further, a commutative ROABP that cannot be efficiently simulated by a diagonal ROABP will give an explicit polynomial that gives a super-polynomial separation between dimension of partial derivatives and Waring rank. Our proof of the above result builds on the results of Marinari, M\"oller and Mora (1993), and M\"oller and Stetter (1995), that characterise rings of commuting matrices in terms of polynomials that have small dimension of partial derivatives. The algebraic structure of the coefficient matrices of these ROABPs plays a crucial role in our proofs.