Abstract
We present a range of new results for testing properties of Boolean functions that are defined in terms of the Fourier spectrum. Broadly speaking, our results show that the property of a Boolean function having a concise Fourier representation is locally testable.
We first give an efficient algorithm for testing whether the Fourier spectrum of a Boolean function is supported in a low-dimensional subspace of \({\mathbb F}_2^n\) (equivalently, for testing whether f is a junta over a small number of parities). We next give an efficient algorithm for testing whether a Boolean function has a sparse Fourier spectrum (small number of nonzero coefficients). In both cases we also prove lower bounds showing that any testing algorithm — even an adaptive one — must have query complexity within a polynomial factor of our algorithms, which are nonadaptive. Finally, we give an “implicit learning” algorithm that lets us test any sub-property of Fourier concision.
Our technical contributions include new structural results about sparse Boolean functions and new analysis of the pairwise independent hashing of Fourier coefficients from [12].
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References
Alon, N., Fischer, E., Newman, I., Shapira, A.: A combinatorial characterization of the testable graph properties: It’s all about regularity. In: Proc. STOC (2006)
Alon, N., Kaufman, T., Krivelevich, M., Litsyn, S., Ron, D.: Testing low-degree polynomials over GF(2). In: Arora, S., Jansen, K., Rolim, J.D.P., Sahai, A. (eds.) RANDOM 2003. LNCS, vol. 2764, pp. 188–199. Springer, Heidelberg (2003)
Alon, N., Shapira, A.: A characterization of the (natural) graph properties testable with one-sided error. In: Proc. FOCS, pp. 429–438 (2005)
Alon, N., Shapira, A.: Every monotone graph property is testable. In: Proc. STOC, pp. 128–137 (2005)
Austin, T., Tao, T.: On the testability and repair of hereditary hypergraph properties. Random Structures and Algorithms (submitted, 2008)
Bellare, M., Coppersmith, D., Hastad, J., Kiwi, M., Sudan, M.: Linearity testing in characteristic two. IEEE Trans. on Information Theory 42(6), 1781–1795 (1996)
Bellare, M., Goldreich, O., Sudan, M.: Free bits, pcps and non-approximability-towards tight results. SIAM J. Comput. 27(3), 804–915 (1998)
Bernasconi, A., Codenotti, B.: Spectral analysis of boolean functions as a graph eigenvalue problem. IEEE Trans. Computers 48(3), 345–351 (1999)
Blais, E.: Improved bounds for testing juntas. In: Goel, A., Jansen, K., Rolim, J.D.P., Rubinfeld, R. (eds.) APPROX and RANDOM 2008. LNCS, vol. 5171, pp. 317–330. Springer, Heidelberg (2008)
Blum, M., Luby, M., Rubinfeld, R.: Self-testing/correcting with applications to numerical problems. J. Comp. Sys. Sci. 47, 549–595 (1993); Earlier version in STOC 1990
Diakonikolas, I., Lee, H., Matulef, K., Onak, K., Rubinfeld, R., Servedio, R., Wan, A.: Testing for concise representations. In: Proc. FOCS, pp. 549–558 (2007)
Feldman, V., Gopalan, P., Khot, S., Ponnuswami, A.: New results for learning noisy parities and halfspaces. In: Proc. FOCS, pp. 563–576 (2006)
Fischer, E.: The art of uninformed decisions: A primer to property testing. Bulletin of the European Association for Theoretical Computer Science 75, 97–126 (2001)
Fischer, E., Kindler, G., Ron, D., Safra, S., Samorodnitsky, A.: Testing juntas. J. Computer & System Sciences 68(4), 753–787 (2004)
Jackson, J.: An efficient membership-query algorithm for learning DNF with respect to the uniform distribution. Journal of Computer and System Sciences 55, 414–440 (1997)
Kaufman, T., Sudan, M.: Sparse random linear codes are locally decodable and testable. In: Proc. FOCS, pp. 590–600 (2007)
Kaufman, T., Sudan, M.: Algebraic property testing: the role of invariance. In: Proc. 40th Annual ACM Symposium on Theory of Computing (STOC), pp. 403–412 (2008)
Kushilevitz, E., Mansour, Y.: Learning decision trees using the fourier spectrum. SIAM Journal on Computing 22(6), 1331–1348 (1993)
Linial, N., Mansour, Y., Nisan, N.: Constant depth circuits, Fourier transform and learnability. Journal of the ACM 40(3), 607–620 (1993)
Matulef, K., O’Donnell, R., Rubinfeld, R., Servedio, R.: Testing halfspaces. In: Proc. SODA, pp. 256–264 (2009)
Parnas, M., Ron, D., Samorodnitsky, A.: Testing basic boolean formulae. SIAM J. Disc. Math. 16, 20–46 (2002)
Samorodnitsky, A.: Low-degree tests at large distances. In: Proc. 39th ACM Symposium on the Theory of Computing (STOC 2007), pp. 506–515 (2007)
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Gopalan, P., O’Donnell, R., Servedio, R.A., Shpilka, A., Wimmer, K. (2009). Testing Fourier Dimensionality and Sparsity. In: Albers, S., Marchetti-Spaccamela, A., Matias, Y., Nikoletseas, S., Thomas, W. (eds) Automata, Languages and Programming. ICALP 2009. Lecture Notes in Computer Science, vol 5555. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-02927-1_42
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