Testing Unateness of Real-Valued Functions

Abstract: We give a unateness tester for functions of the form $f:[n]^d\rightarrow R$, where $n,d\in \mathbb{N}$ and $R\subseteq \mathbb{R}$ with query complexity $O(\frac{d\log (\max(d,n))}{\epsilon})$. Previously known unateness testers work only for Boolean functions over the domain ${0,1}<sup>d$. We show that every unateness tester for real-valued functions over hypergrid has query complexity $\Omega(\min{d, |R|<sup>2})$. Consequently, our tester is nearly optimal for real-valued functions over ${0,1}<sup>d$. We also prove that every nonadaptive, 1-sided error unateness tester for Boolean functions needs $\Omega(\sqrt{d}/\epsilon)$ queries. Previously, no lower bounds for testing unateness were known.