Improved Hitting Set for Orbit of ROABPs

Abstract

The orbit of an n-variate polynomial $f({\rm x})$ over a field $\mathbb{F}$ is the set $\{f(A{\rm x} + {\rm b})\,\mid\, A\in{\rm GL}(n, \mathbb{F})\mbox{ and }{\rm b}\in \mathbb{F}^n\}$, and the orbit of a polynomial class is the union of orbits of the polynomials in it. In this paper, we give improved constructions of hitting sets for the orbit of read-once oblivious algebraic branching programs (ROABPs) and a related model. Over fields with characteristic zero or greater than $d$, we construct a hitting set of size $(ndw)^{O(w^2\log n\cdot \min\{w^2, d\log w\})}$ for the orbit of ROABPs in unknown variable order where $d$ is the individual degree and $w$ is the width of ROABPs. We also give a hitting set of size $(ndw)^{O(\min\{w^2,d\log w\})}$ for the orbit of polynomials computed by width-w ROABPs in any variable order. Our hitting sets improve upon the results of Saha & Thankey (2021) who gave an $(ndw)^{O(d\log w)}$ size hitting set for the orbit of commutative ROABPs (a subclass of any-order ROABPs) and $(nw)^{O(w^6\log n)}$ size hitting set for the orbit of multilinear ROABPs. Designing better hitting sets in large individual degree regime, for instance $d>n$, was asked as an open problem by Saha & Thankey (2021) and this work solves it in small width setting.

We prove some new rank concentration results by establishing low-cone concentration for polynomials over vector spaces, and they strengthen some previously known low-support-based rank concentration results shown in Forbes et al. (2013). These new low-cone concentration results are crucial in our hitting set construction, and may be of independent interest. To the best of our knowledge, this is the first time when low-cone rank concentration has been used for designing hitting sets.