On Approximating the Number of Relevant Variables in a Function

In this work we consider the problem of approximating the number of relevant variables in a function given query access to the function. Since obtaining a multiplicative factor approximation is hard in general, we consider several relaxations of the problem. In particular, we consider a relaxation of the property testing variant of the problem and we consider relaxations in which we have a promise that the function belongs to a certain family of functions (e.g., linear functions). In the former relaxation the task is to distinguish between the case that the number of relevant variables is at most k, and the case in which it is far from any function in which the number of relevant variables is more than (1 + γ)k for a parameter γ. We give both upper bounds and almost matching lower bounds for the relaxations we study.

[1]  Ronitt Rubinfeld,et al.  Approximating the Influence of Monotone Boolean Functions in $O(\sqrt{n})$ Query Complexity , 2011, APPROX-RANDOM.

[2]  Joshua Brody,et al.  Property Testing Lower Bounds via Communication Complexity , 2011, 2011 IEEE 26th Annual Conference on Computational Complexity.

[3]  Gregory Valiant,et al.  Estimating the unseen: an n/log(n)-sample estimator for entropy and support size, shown optimal via new CLTs , 2011, STOC '11.

[4]  Ronitt Rubinfeld,et al.  Approximating the Influence of Monotone Boolean Functions in $O(\sqrt{n})$ Query Complexity , 2011, APPROX-RANDOM.

[5]  Paul Valiant Testing symmetric properties of distributions , 2008, STOC '08.

[6]  Oded Goldreich On Testing Computability by Small Width OBDDs , 2010, Electron. Colloquium Comput. Complex..

[7]  Gregory Valiant,et al.  A CLT and tight lower bounds for estimating entropy , 2010, Electron. Colloquium Comput. Complex..

[8]  Gregory Valiant,et al.  Estimating the unseen: A sublinear-sample canonical estimator of distributions , 2010, Electron. Colloquium Comput. Complex..

[9]  Eric Blais Testing juntas nearly optimally , 2009, STOC '09.

[10]  Eric Blais Improved Bounds for Testing Juntas , 2008, APPROX-RANDOM.

[11]  Dana Ron,et al.  Strong Lower Bounds for Approximating Distribution Support Size and the Distinct Elements Problem , 2007, 48th Annual IEEE Symposium on Foundations of Computer Science (FOCS'07).

[12]  Hana Chockler,et al.  A lower bound for testing juntas , 2004, Inf. Process. Lett..

[13]  Guy Kindler,et al.  Testing juntas , 2002, J. Comput. Syst. Sci..

[14]  Private Communications , 2001 .

[15]  Dana Ron,et al.  Testing Problems with Sublearning Sample Complexity , 2000, J. Comput. Syst. Sci..

[16]  Noga Alon,et al.  Testing of clustering , 2000, Proceedings 41st Annual Symposium on Foundations of Computer Science.

[17]  Dana Ron,et al.  Testing the diameter of graphs , 1999, RANDOM-APPROX.

[18]  Dana Ron,et al.  Testing problems with sub-learning sample complexity , 1998, COLT' 98.

[19]  Dana Ron,et al.  Property testing and its connection to learning and approximation , 1996, Proceedings of 37th Conference on Foundations of Computer Science.

[20]  Ronitt Rubinfeld,et al.  Robust Characterizations of Polynomials with Applications to Program Testing , 1996, SIAM J. Comput..

[21]  Ingo Wegener The critical complexity of all (monotone) Boolean functions and monotone graph properties , 1985, FCT.