39 research outputs found

    An ~O(n) Queries Adaptive Tester for Unateness

    Get PDF
    We present an adaptive tester for the unateness property of Boolean functions. Given a function f:{0,1}^n -> {0,1} the tester makes O(n log(n)/epsilon) adaptive queries to the function. The tester always accepts a unate function, and rejects with probability at least 0.9 if a function is epsilon-far from being unate

    Optimal Unateness Testers for Real-Valued Functions: Adaptivity Helps

    Get PDF
    We study the problem of testing unateness of functions f:{0,1}^d -> R. We give an O(d/epsilon . log(d/epsilon))-query nonadaptive tester and an O(d/epsilon)-query adaptive tester and show that both testers are optimal for a fixed distance parameter epsilon. 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. We generalize our results to obtain optimal unateness testers for functions f:[n]^d -> R. Our 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

    Lower Bounds for Tolerant Junta and Unateness Testing via Rejection Sampling of Graphs

    Get PDF
    We introduce a new model for testing graph properties which we call the rejection sampling model. We show that testing bipartiteness of n-nodes graphs using rejection sampling queries requires complexity Omega~(n^2). Via reductions from the rejection sampling model, we give three new lower bounds for tolerant testing of Boolean functions of the form f : {0,1}^n -> {0,1}: - Tolerant k-junta testing with non-adaptive queries requires Omega~(k^2) queries. - Tolerant unateness testing requires Omega~(n) queries. - Tolerant unateness testing with non-adaptive queries requires Omega~(n^{3/2}) queries. Given the O~(k^{3/2})-query non-adaptive junta tester of Blais [Eric Blais, 2008], we conclude that non-adaptive tolerant junta testing requires more queries than non-tolerant junta testing. In addition, given the O~(n^{3/4})-query unateness tester of Chen, Waingarten, and Xie [Xi Chen et al., 2017] and the O~(n)-query non-adaptive unateness tester of Baleshzar, Chakrabarty, Pallavoor, Raskhodnikova, and Seshadhri [Roksana Baleshzar et al., 2017], we conclude that tolerant unateness testing requires more queries than non-tolerant unateness testing, in both adaptive and non-adaptive settings. These lower bounds provide the first separation between tolerant and non-tolerant testing for a natural property of Boolean functions

    Mildly Exponential Lower Bounds on Tolerant Testers for Monotonicity, Unateness, and Juntas

    Full text link
    We give the first super-polynomial (in fact, mildly exponential) lower bounds for tolerant testing (equivalently, distance estimation) of monotonicity, unateness, and juntas with a constant separation between the "yes" and "no" cases. Specifically, we give ∙\bullet A 2Ω(n1/4/ε)2^{\Omega(n^{1/4}/\sqrt{\varepsilon})}-query lower bound for non-adaptive, two-sided tolerant monotonicity testers and unateness testers when the "gap" parameter ε2−ε1\varepsilon_2-\varepsilon_1 is equal to ε\varepsilon, for any ε≥1/n\varepsilon \geq 1/\sqrt{n}; ∙\bullet A 2Ω(k1/2)2^{\Omega(k^{1/2})}-query lower bound for non-adaptive, two-sided tolerant junta testers when the gap parameter is an absolute constant. In the constant-gap regime no non-trivial prior lower bound was known for monotonicity, the best prior lower bound known for unateness was Ω~(n3/2)\tilde{\Omega}(n^{3/2}) queries, and the best prior lower bound known for juntas was poly(k)\mathrm{poly}(k) queries.Comment: 20 pages, 1 figur

    Almost Optimal Testers for Concise Representations

    Get PDF
    We give improved and almost optimal testers for several classes of Boolean functions on n variables that have concise representation in the uniform and distribution-free model. Classes, such as k-Junta, k-Linear, s-Term DNF, s-Term Monotone DNF, r-DNF, Decision List, r-Decision List, size-s Decision Tree, size-s Boolean Formula, size-s Branching Program, s-Sparse Polynomial over the binary field and functions with Fourier Degree at most d. The approach is new and combines ideas from Diakonikolas et al. [Ilias Diakonikolas et al., 2007], Bshouty [Nader H. Bshouty, 2018], Goldreich et al. [Oded Goldreich et al., 1998], and learning theory. The method can be extended to several other classes of functions over any domain that can be approximated by functions with a small number of relevant variables
    corecore