Abstract
For a test T ⊆ {0, 1}n, define k*(T) to be the maximum k such that there exists a k-wise uniform distribution over {0, 1}n whose support is a subset of T.
For Ht = {x ∈ {0, 1}n : | ∑ixi − n/2| ≤ t}, we prove k*(Ht) = Θ (t2/n + 1).
For Sm, c = {x ∈ {0, 1}n : ∑ixi ≡ c (mod m)}, we prove that k*(Sm, c) = Θ (n/m2). For some k = O(n/m) we also show that any k-wise uniform distribution puts probability mass at most 1/m + 1/100 over Sm, c. Finally, for any fixed odd m we show that there is an integer k = (1 − Ω(1))n such that any k-wise uniform distribution lands in T with probability exponentially close to |Sm, c|/2n; and this result is false for any even m.
- Miklos Ajtai and Avi Wigderson. 1989. Deterministic simulation of probabilistic constant-depth circuits. Advances in Computing Research—Randomness and Computation 5 (1989), 199--223.Google Scholar
- Noga Alon, Oded Goldreich, and Yishay Mansour. 2003. Almost k-wise independence versus k-wise independence. Inf. Process. Lett. 88, 3 (2003), 107--110. Google Scholar
Digital Library
- Louay M. J. Bazzi. 2009. Polylogarithmic independence can fool DNF formulas. SIAM J. Comput. 38, 6 (2009), 2220--2272. Google Scholar
Digital Library
- Avraham Ben-Aroya, Dean Doron, and Amnon Ta-Shma. 2017. An efficient reduction from two-source to non-malleable extractors: Achieving near-logarithmic min-entropy. In Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing (STOC’17). ACM, New York, 1185--1194. Google Scholar
Digital Library
- Ravi Boppana, Johan Håstad, Chin Ho Lee, and Emanuele Viola. 2016. Bounded independence vs. moduli. In Proceedings of the 20th International Workshop on Randomization and Computation (RANDOM’16), Leibniz International Proceedings in Informatics, Vol. 60. Schloss Dagstuhl, 24:1--24:9.Google Scholar
- Mark Braverman. 2010. Polylogarithmic independence fools AC<sup>0</sup> circuits. J. ACM 57, 5 (2010). Google Scholar
Digital Library
- Neal Carothers. {n.d.}. A Short Course on Approximation Theory. Retrieved from http://fourier.math.uoc.gr/∼mk/approx1011/carothers.pdf.Google Scholar
- J. Lawrence Carter and Mark N. Wegman. 1979. Universal classes of hash functions. J. Comput. Syst. Sci. 18, 2 (1979), 143--154.Google Scholar
Cross Ref
- Suresh Chari, Pankaj Rohatgi, and Aravind Srinivasan. 2000. Improved algorithms via approximations of probability distributions. J. Comput. Syste. Sci. 61, 1 (2000), 81--107. Google Scholar
Digital Library
- Eshan Chattopadhyay and David Zuckerman. 2016. Explicit two-source extractors and resilient functions. In Proceedings of the 48th ACM Symposium on the Theory of Computing (STOC’16). 670--683. Google Scholar
Digital Library
- Elliott Cheney. 1966. Introduction to Approximation Theory. McGraw-Hill, New York, NY.Google Scholar
- Ilias Diakonikolas, Parikshit Gopalan, Ragesh Jaiswal, Rocco A. Servedio, and Emanuele Viola. 2010. Bounded independence fools halfspaces. SIAM J. Comput. 39, 8 (2010), 3441--3462. Google Scholar
Digital Library
- Ilias Diakonikolas, Daniel Kane, and Jelani Nelson. 2010. Bounded independence fools degree-2 threshold functions. In Proceedings of the 51st IEEE Symposium on Foundations of Computer Science (FOCS’10). IEEE, 11--20. Google Scholar
Digital Library
- Guy Even, Oded Goldreich, Michael Luby, Noam Nisan, and Boban Velickovic. 1992. Approximations of general independent distributions. In Proceedings of the 24th ACM Symposium on the Theory of Computing (STOC’92). 10--16. Google Scholar
Digital Library
- William Feller. 1971. An Introduction to Probability Theory and Its Applications (2nd ed.). Vol. 2. Wiley.Google Scholar
- Parikshit Gopalan, Raghu Meka, Omer Reingold, Luca Trevisan, and Salil Vadhan. 2012. Better pseudorandom generators from milder pseudorandom restrictions. In Proceedings of the 53rd IEEE Symposium on Foundations of Computer Science (FOCS’12). 120--129. Google Scholar
Digital Library
- Parikshit Gopalan, Ryan O’Donnell, Yi Wu, and David Zuckerman. 2010. Fooling functions of halfspaces under product distributions. In Proceedings of the 25th IEEE Conference on Computational Complexity (CCC’10). IEEE, 223--234. Google Scholar
Digital Library
- Uffe Haagerup. 1981. The best constants in the Khintchine inequality. Stud. Math. 70, 3 (1981), 231--283.Google Scholar
Cross Ref
- Elad Haramaty, Chin Ho Lee, and Emanuele Viola. 2018. Bounded independence plus noise fools products. SIAM J. Comput. 47, 2 (2018), 493--523.Google Scholar
Cross Ref
- Prahladh Harsha and Srikanth Srinivasan. 2016. On polynomial approximations to AC<sup>0</sup>. In Proceedings of the 20th International Workshop on Randomization and Computation (RANDOM’16), Leibniz International Proceedings in Informatics, Vol. 60. Schloss Dagstuhl, 32:1--32:14.Google Scholar
- Aleksandr Khintchine. 1923. Über dyadische Brüche. Math. Zeitschr. 18, 1 (1923), 109--116.Google Scholar
Cross Ref
- Chin Ho Lee and Emanuele Viola. 2017. More on bounded independence plus noise: Pseudorandom generators for read-once polynomials. Retreived from http://www.ccs.neu.edu/home/viola/.Google Scholar
- Chin Ho Lee and Emanuele Viola. 2017. Some limitations of the sum of small-bias distributions. Theory Comput. 13, Article 16 (2017), 23 pages.Google Scholar
- Raghu Meka and David Zuckerman. 2009. Small-bias spaces for group products. In Proceedings of the 13th Workshop on Randomization and Computation (RANDOM’09), Lecture Notes in Computer Science, Vol. 5687. Springer, 658--672. Google Scholar
Digital Library
- Ryan O’Donnell. 2014. Analysis of Boolean Functions. Cambridge University Press. Google Scholar
Digital Library
- Yuval Rabani and Amir Shpilka. 2010. Explicit construction of a small epsilon-net for linear threshold functions. SIAM J. Comput. 39, 8 (2010), 3501--3520. Google Scholar
Digital Library
- Alexander A. Razborov. 1987. Lower bounds on the dimension of schemes of bounded depth in a complete basis containing the logical addition function. Akad. Nauk SSSR. Mat. Zamet. 41, 4 (1987), 598--607.Google Scholar
- Alexander A. Razborov. 2009. A simple proof of Bazzi’s theorem. ACM Trans. Comput. Theory 1, 1 (2009). https://dl.acm.org/citation.cfm?id=1490273. Google Scholar
Digital Library
- Herbert Robbins. 1955. A remark on Stirling’s formula. Am. Math. Month. 62, 1 (Jan. 1955), 26--29.Google Scholar
Cross Ref
- Roman Smolensky. 1987. Algebraic methods in the theory of lower bounds for Boolean circuit complexity. In Proceedings of the 19th ACM Symposium on the Theory of Computing (STOC’87). ACM, 77--82. Google Scholar
Digital Library
- Avishay Tal. 2017. Tight bounds on the Fourier spectrum of AC<sup>0</sup>. In Proceedings of the 32nd Computational Complexity Conference. LIPIcs. Leibniz Int. Proc. Inform., Vol. 79. Schloss Dagstuhl. Leibniz-Zent. Inform., Wadern, 15:1--15:31. Google Scholar
Digital Library
- Emanuele Viola. 2017. Special topics in complexity theory. Lecture notes of the class taught at Northeastern University. Retrieved from http://www.ccs.neu.edu/home/viola/classes/spepf17.html.Google Scholar
- Emanuele Viola and Avi Wigderson. 2008. Norms, XOR lemmas, and lower bounds for polynomials and protocols. Theory Comput. 4 (2008), 137--168.Google Scholar
Cross Ref
Index Terms
Bounded Independence versus Symmetric Tests
Recommendations
Almost k-wise independence versus k-wise independence
We say that a distribution over {0,1}n is (ε,k)-wise independent if its restriction to every k coordinates results in a distribution that is ε-close to the uniform distribution. A natural question regarding (ε, k)-wise independent distributions is how ...
Bounded independence plus noise fools products
CCC '17: Proceedings of the 32nd Computational Complexity ConferenceLet D be a b-wise independent distribution over {0, 1}m. Let E be the "noise" distribution over {0, 1}m where the bits are independent and each bit is 1 with probability η/2. We study which tests f: {0, 1}m → [−1, 1] are ε-fooled by D + E, i.e., | E[f(D ...
Improved lower bound on the number of balanced symmetric functions over GF(p)
The lower bound on the number of n-variable balanced symmetric functions over finite fields GF ( p ) presented by Cusick et al. in T.W. Cusick, Y. Li, P. Sta nica , Balanced symmetric functions over GF(p), IEEE Trans. Inform. Theory 54 (3) (2008) 1304-...






Comments