Testing Unateness of Real-Valued Functions

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