Testing Properties of Linear Functions
نویسندگان
چکیده
The function f : F2 → F2 is k-linear if it returns the sum (over F2) of exactly k coordinates of its input. We introduce strong lower bounds on the query complexity for testing whether a function is k-linear. We show that for any k ≤ n 2 , at least k−o(k) queries are required to test k-linearity, and we show that when k ≈ n 2 , this lower bound is nearly tight since 4 3 k+o(k) queries are sufficient to test k-linearity. We also show that non-adaptive testers require 2k − o(k) queries to test k-linearity. Our results improve known lower bounds on the query complexity of a number of property testing problems: juntas, sparse polynomials, small decision trees, and functions with low Fourier degree. Our lower bounds also give new results in testing function isomorphism and the characterization of the query complexity for testing properties of linear functions.
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