Property Testing pp 228-233

Testing Monotone Continuous Distributions on High-Dimensional Real Cubes

• Artur Czumaj
• Christian Sohler
Chapter
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6390)

Abstract

We study the task of testing properties of probability distributions and our focus is on understanding the role of continuous distributions in this setting. We consider a scenario in which we have access to independent samples of an unknown distribution $$\mathfrak{D}$$ with infinite (perhaps even uncountable) support. Our goal is to test whether $$\mathfrak{D}$$ has a given property or it is ε-far from it (in the statistical distance, with the L 1-distance measure).

It is not difficult to see that for many natural distributions on infinite or uncountable domains, no algorithm can exist and the central objective of our study is to understand if there are any nontrivial distributions that can be efficiently tested. For example, it is easy to see that there is no algorithm that tests if a given probability distribution on [0,1] is uniform. We show however, that if some additional information about the input distribution is known, testing uniform distribution is possible. We extend the recent result about testing uniformity for monotone distributions on Boolean n-dimensional cubes by Rubinfeld and Servedio (STOC’2005) to the case of continuous [0,1] n cubes. We show that if a distribution $$\mathfrak{D}$$ on [0,1] n is monotone, then one can test if $$\mathfrak{D}$$ is uniform with the sample complexity $$\mathcal{O}(n/\epsilon^2)$$. This result is optimal up to a polylogarithmic factor. We also extend the result of Rubinfeld and Servedio (STOC’2005) to test if a distribution $$\mathfrak{D}$$ on {0,1,...,k} n is monotone with the sample complexity $$\mathcal{O}(n/\epsilon^2)$$.

Keywords

testing distributions

References

1. 1.
Adamaszek, M., Czumaj, A., Sohler, C.: Testing monotone continuous distributions on high-dimensional real cubes. In: Proc. 21st Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 56–65 (2010)Google Scholar
2. 2.
Alon, N., Andoni, A., Kaufman, T., Matulef, K., Rubinfeld, R., Xie, N.: Testing k-wise and almost k-wise Independence. In: Proc. 39th Annual ACM Symposium on Theory of Computing, pp. 496–505 (2007)Google Scholar
3. 3.
Batu, T., Dasgupta, S., Kumar, R., Rubinfeld, R.: The complexity of approximating the entropy. In: Proc. 34th Annual ACM Symposium on Theory of Computing, pp. 678–687 (2002)Google Scholar
4. 4.
Batu, T., Fischer, E., Fortnow, L., Kumar, R., Rubinfeld, R., White, P.: Testing random variables for independence and identity. In: Proc. 42nd IEEE Symposium on Foundations of Computer Science, pp. 442–415 (2001)Google Scholar
5. 5.
Batu, T., Fortnow, L., Rubinfeld, R., Smith, W.D., White, P.: Testing that distributions are close. In: Proc. 41st IEEE Symposium on Foundations of Computer Science, pp. 259–269 (2000)Google Scholar
6. 6.
Batu, T., Kumar, R., Rubinfeld, R.: Sublinear algorithms for testing monotone and unimodal distributions. In: Proc. 36th Annual ACM Symposium on Theory of Computing, pp. 381–390 (2004)Google Scholar
7. 7.
Goldreich, O., Ron, D.: On testing expansion in bounded-degree graphs. Electronic Colloquium on Computational Complexity, Report No. 7 (2000)Google Scholar
8. 8.
Raskhodnikova, S., Ron, D., Shpilka, A., Smith, A.: Strong lower bounds for approximating distribution support size and the distinct elements problem. SIAM Journal on Computing 39(3), 813–842 (2009)
9. 9.
Rubinfeld, R.: Sublinear time algorithms. In: Proc. International Congress of Mathematicians, Madrid, Spain, August 22-30 (2006)Google Scholar
10. 10.
Rubinfeld, R., Servedio, R.A.: Testing monotone high-dimensional distributions. In: Proc. 37th Annual ACM Symposium on Theory of Computing, pp. 147–156 (2005)Google Scholar
11. 11.
Valiant, P.: Testing symmetric properties of distributions. In: Proc. 40th Annual ACM Symposium on Theory of Computing, pp. 383–392 (2008)Google Scholar