Testing Unateness of Real-Valued Functions

We give a unateness tester for functions of the form f : [n] → R, where n, d ∈ N and R ⊆ R with query complexity O( log(max(d,n)) ). Previously known unateness testers work only for Boolean functions over the domain {0, 1}. We show that every unateness tester for realvalued functions over hypergrid has query complexity Ω(min{d, |R|}). Consequently, our tester is nearly optimal for real-valued functions over {0, 1}. We also prove that every nonadaptive, 1-sided error unateness tester for Boolean functions needs Ω( √ d/ ) queries. Previously, no lower bounds for testing unateness were known.