Abstract
A function defined on the Boolean hypercube is k-Fourier-sparse if it has at most k nonzero Fourier coefficients. For a function f: F2n → R and parameters k and d, we prove a strong upper bound on the number of k-Fourier-sparse Boolean functions that disagree with f on at most d inputs. Our bound implies that the number of uniform and independent random samples needed for learning the class of k-Fourier-sparse Boolean functions on n variables exactly is at most O(n · klog k).
As an application, we prove an upper bound on the query complexity of testing Booleanity of Fourier-sparse functions. Our bound is tight up to a logarithmic factor and quadratically improves on a result due to Gur and Tamuz [2013].
- Noga Alon, Tali Kaufman, Michael Krivelevich, Simon Litsyn, and Dana Ron. 2005. Testing Reed-Muller codes. IEEE Trans. Inf. Theory 51, 11 (2005), 4032--4039. Preliminary version in RANDOM’03. Google Scholar
Digital Library
- Alexandr Andoni, Rina Panigrahy, Gregory Valiant, and Li Zhang. 2014. Learning sparse polynomial functions. In SODA. 500--510. Google Scholar
Digital Library
- Khanh Do Ba, Piotr Indyk, Eric Price, and David P. Woodruff. 2010. Lower bounds for sparse recovery. In SODA. 1190--1197. Google Scholar
Digital Library
- Anna Bernasconi and Bruno Codenotti. 1999. Spectral analysis of Boolean functions as a graph eigenvalue problem. IEEE Trans. Comput. 48, 3 (1999), 345--351. Google Scholar
Digital Library
- Arnab Bhattacharyya. 2013. Guest column: On testing affine-invariant properties over finite fields. SIGACT News 44, 4 (2013), 53--72. Google Scholar
Digital Library
- Arnab Bhattacharyya, Swastik Kopparty, Grant Schoenebeck, Madhu Sudan, and David Zuckerman. 2010. Optimal testing of Reed-Muller codes. In FOCS. 488--497. Google Scholar
Digital Library
- Avrim Blum. 2003. Learning a function of r relevant variables. In COLT. 731--733.Google Scholar
- Jean Bourgain. 2014. An improved estimate in the restricted isometry problem. In Geometric Aspects of Functional Analysis. Lecture Notes in Mathematics, Vol. 2116. Springer, 65--70.Google Scholar
- Jehoshua Bruck and Roman Smolensky. 1992. Polynomial threshold functions, AC0 functions, and spectral norms. SIAM J. Comput. 21, 1 (1992), 33--42. Preliminary version in FOCS’90. Google Scholar
Digital Library
- Emmanuel J. Candès and Terence Tao. 2006. Near-optimal signal recovery from random projections: Universal encoding strategies? IEEE Trans. Inf. Theory 52, 12 (2006), 5406--5425. Google Scholar
Digital Library
- Mahdi Cheraghchi, Venkatesan Guruswami, and Ameya Velingker. 2013. Restricted isometry of Fourier matrices and list decodability of random linear codes. SIAM J. Comput. 42, 5 (2013), 1888--1914. Preliminary version in SODA’13.Google Scholar
Cross Ref
- Oded Goldreich and Leonid A. Levin. 1989. A hard-core predicate for all one-way functions. In STOC. 25--32. Google Scholar
Digital Library
- Parikshit Gopalan, Ryan O’Donnell, Rocco A. Servedio, Amir Shpilka, and Karl Wimmer. 2011. Testing Fourier dimensionality and sparsity. SIAM J. Comput. 40, 4 (2011), 1075--1100. Preliminary version in ICALP’09. Google Scholar
Digital Library
- Tom Gur and Omer Tamuz. 2013. Testing Booleanity and the uncertainty principle. Chicago J. Theor. Comput. Sci. 2013 (2013).Google Scholar
- Ishay Haviv and Oded Regev. 2016. The restricted isometry property of subsampled Fourier matrices. In SODA. 288--297. Google Scholar
Digital Library
- Stasys Jukna. 2011. Extremal Combinatorics: With Applications in Computer Science (2nd ed.). Springer-Verlag. Google Scholar
Digital Library
- Tali Kaufman, Shachar Lovett, and Ely Porat. 2012. Weight distribution and list-decoding size of Reed-Muller codes. IEEE Trans. Inf. Theory 58, 5 (2012), 2689--2696. Preliminary version in ICS’10. Google Scholar
Digital Library
- Murat Kocaoglu, Karthikeyan Shanmugam, Alexandros G. Dimakis, and Adam R. Klivans. 2014. Sparse polynomial learning and graph sketching. In NIPS. 3122--3130.Google Scholar
- Eyal Kushilevitz and Yishay Mansour. 1993. Learning decision trees using the Fourier spectrum. SIAM J. Comput. 22, 6 (1993), 1331--1348. Preliminary version in STOC’91. Google Scholar
Digital Library
- Nathan Linial, Yishay Mansour, and Noam Nisan. 1993. Constant depth circuits, Fourier transform, and learnability. J. ACM 40, 3 (1993), 607--620. Preliminary version in FOCS’89. Google Scholar
Digital Library
- Elchanan Mossel, Ryan O’Donnell, and Rocco A. Servedio. 2004. Learning functions of k relevant variables. J. Comput. Syst. Sci. 69, 3 (2004), 421--434. Preliminary vesion in STOC’03. Google Scholar
Digital Library
- Ryan O’Donnell. 2014. Analysis of Boolean Functions. Cambridge University Press. Google Scholar
Digital Library
- Mark Rudelson and Roman Vershynin. 2008. On sparse reconstruction from Fourier and Gaussian measurements. Comm. Pure Appl. Math. 61, 8 (2008), 1025--1045.Google Scholar
Cross Ref
- Madhu Sudan. 2010. Invariance in property testing. In Property Testing - Current Research and Surveys, Vol. 6390. Springer, 211--227. Google Scholar
Digital Library
- Gregory Valiant. 2012. Finding correlations in subquadratic time, with applications to learning parities and juntas. In FOCS. 11--20. Google Scholar
Digital Library
Index Terms
The List-Decoding Size of Fourier-Sparse Boolean Functions
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