We develop a new technique for proving lower bounds in property testing, by showing a strong connection between testing and communication complexity. We give a simple scheme for reducing communication problems to testing problems, thus allowing us to use known lower bounds in communication complexity to prove lower bounds in testing. This scheme is general and implies a number of new testing bounds, as well as simpler proofs of several known bounds.For the problem of testing whether a boolean function is k-linear (a parity function on k variables), we achieve a lower bound of Ω(k) queries, even for adaptive algorithms with two-sided error, thus confirming a conjecture of Goldreich [25]. The same argument behind this lower bound also implies a new proof of known lower bounds for testing related classes such as k-juntas. For some classes, such as the class of monotone functions and the class of s-sparse GF(2) polynomials, we significantly strengthen the best known bounds.
This paper addresses the problem of testing whether a Boolean-valued function f is a halfspace, i.e. a function of the form f (x) = sgn(w · x − θ). We consider halfspaces over the continuous domain R n (endowed with the standard multivariate Gaussian distribution) as well as halfspaces over the Boolean cube {−1, 1} n (endowed with the uniform distribution). In both cases we give an algorithm that distinguishes halfspaces from functions that are -far from any halfspace using only poly( 1 ) queries, independent of the dimension n.Two simple structural results about halfspaces are at the heart of our approach for the Gaussian distribution: the first gives an exact relationship between the expected value of a halfspace f and the sum of the squares of f 's degree-1 Hermite coefficients, and the second shows that any function that approximately satisfies this relationship is close to a halfspace. We prove analogous results for the Boolean cube {−1, 1} n (with Fourier coefficients in place of Hermite coefficients) for balanced halfspaces in which all degree-1 Fourier coefficients are small. Dealing with general halfspaces over {−1, 1} n poses significant additional complications and requires other ingredients. These include "cross-consistency" versions of the results mentioned above for pairs of halfspaces with the same weights but different thresholds; new structural results relating the largest degree-1 Fourier coefficient and the largest weight in unbalanced halfspaces; and algorithmic techniques from recent work on testing juntas [FKR + 02].
This paper addresses the problem of testing whether a Boolean-valued function f is a halfspace, i.e. a function of the form f (x) = sgn(w · x − θ). We consider halfspaces over the continuous domain R n (endowed with the standard multivariate Gaussian distribution) as well as halfspaces over the Boolean cube {−1, 1} n (endowed with the uniform distribution). In both cases we give an algorithm that distinguishes halfspaces from functions that are ǫ-far from any halfspace using only poly( 1 ǫ ) queries, independent of the dimension n.Two simple structural results about halfspaces are at the heart of our approach for the Gaussian distribution: the first gives an exact relationship between the expected value of a halfspace f and the sum of the squares of f 's degree-1 Hermite coefficients, and the second shows that any function that approximately satisfies this relationship is close to a halfspace. We prove analogous results for the Boolean cube {−1, 1} n (with Fourier coefficients in place of Hermite coefficients) for balanced halfspaces in which all degree-1 Fourier coefficients are small. Dealing with general halfspaces over {−1, 1} n poses significant additional complications and requires other ingredients. These include "cross-consistency" versions of the results mentioned above for pairs of halfspaces with the same weights but different thresholds; new structural results relating the largest degree-1 Fourier coefficient and the largest weight in unbalanced halfspaces; and algorithmic techniques from recent work on testing juntas [FKR + 02].
In this work, we consider the problems of testing whether a distribution over {0, 1} n is k-wise (resp. ( , k)-wise) independent using samples drawn from that distribution.For the problem of distinguishing k-wise independent distributions from those that are δ-far from k-wise independence in statistical distance, we upper bound the number of required samples byÕ(n k /δ 2 ) and lower bound it by Ω(n k−1 2 /δ) (these bounds hold for constant k, and essentially the same bounds hold for general k). To achieve these bounds, we use Fourier analysis to relate a distribution's distance from k-wise independence to its biases, a measure of the parity imbalance it induces on a set of variables. The relationships we derive are tighter than previously known, and may be of independent interest.To distinguish ( , k)-wise independent distributions from those that are δ-far from ( , k)-wise independence in statistical distance, we upper bound the number of required samples by O`k log n δ 2 2´a nd lower bound it by Ω " √ k log n 2 k ( +δ) √ log 1/2 k ( +δ) «. Although these bounds are an exponential improvement (in terms of n and k) over the corresponding bounds for testing k-wise independence, we give evidence that the time complexity of testing ( , k)-wise independence is unlikely to be poly(n, 1/ , 1/δ) for k = Θ(log n), since this would disprove a plausible conjecture concerning the hardness of finding hidden cliques in random graphs. Under the conjecture, our result implies that for, say, k = log n and = 1/n 0.99 , there is a set of ( , k)-wise indepen-
We describe a general method for testing whether a function on n input variables has a concise representation. The approach combines ideas from the junta test of Fischer et al. [FKR + 04] with ideas from learning theory, and yields property testers that make poly(s/ǫ) queries (independent of n) for Boolean function classes such as s-term DNF formulas (answering a question posed by Parnas et al.[PRS02]), size-s decision trees, size-s Boolean formulas, and size-s Boolean circuits.The method can be applied to non-Boolean valued function classes as well. This is achieved via a generalization of the notion of variation from Fischer et al. to non-Boolean functions. Using this generalization we extend the original junta test of Fischer et al. to work for non-Boolean functions, and give poly(s/ǫ)-query testing algorithms for non-Boolean valued function classes such as size-s algebraic circuits and s-sparse polynomials over finite fields.We also prove aΩ( √ s) query lower bound for nonadaptively testing s-sparse polynomials over finite fields of constant size. This shows that in some instances, our general method yields a property tester with query complexity that is optimal (for nonadaptive algorithms) up to a polynomial factor.
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