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

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We study the problem of testing unateness of functions f : {0, 1} d → R. A function f : {0, 1} d → R is unate if for every coordinate i ∈ [d], the function is either nonincreasing in the i th coordinate or nondecreasing in the i th coordinate. We give an O((d/ε) • log(d/ε))-query nonadaptive tester and an O(d/ε)-query adaptive tester and show that both testers are optimal for a fixed distance parameter ε. Previously known unateness testers worked only for Boolean functions, and their query complexity had worse dependence on the dimension d both for the adaptive and the nonadaptive case. Moreover, no lower bounds for testing unateness were known. (Concurrent work by Chen et al. (STOC'17) proved an Ω(d/ log 2 d) lower bound on the nonadaptive query complexity of testing unateness of Boolean functions.) We also generalize our results to obtain optimal unateness testers for functions f : [n] d → R.

We study the problem of testing unateness of functions f : {0, 1} d → R. We give a O( d ε •log d ε )query nonadaptive tester and a O( d ε )-query adaptive tester and show that both testers are optimal for a fixed distance parameter ε. Previously known unateness testers worked only for Boolean functions, and their query complexity had worse dependence on the dimension both for the adaptive and the nonadaptive case. Moreover, no lower bounds for testing unateness were known 1 . We also generalize our results to obtain optimal unateness testers for functionsOur results establish that adaptivity helps with testing unateness of real-valued functions on domains of the form {0, 1} d and, more generally, [n] d . This stands in contrast to the situation for monotonicity testing where there is no adaptivity gap for functions f : [n] d → R. * Preliminary (much weaker) versions of this paper were posted as [16] and [2].