Efficiently Testing Sparse GF(2) Polynomials.
I. Diakonikolas and H. Lee and K. Matulef and R. Servedio and A. Wan.
Algorithmica, 61(3), 2011, pp. 580-605.
Preliminary version in 35th International Conference on Automata, Languages and Programming (ICALP), 2008, pp. 502--514.

Abstract:

We give the first algorithm that is both query-efficient and time-efficient for testing whether an unknown function $f: \{0,1\}^n \to \{-1,1\}$ is an $s$-sparse $GF(2)$ polynomial versus $\eps$-far from every such polynomial. Our algorithm makes $\poly(s,1/\eps)$ black-box queries to $f$ and runs in time $n \cdot \poly(s,1/\eps)$. The only previous algorithm for this testing problem \cite{DLM+:07} used poly$(s,1/\eps)$ queries, but had running time exponential in $s$ and super-polynomial in $1/\eps$.

Our approach significantly extends the testing by implicit learning'' methodology of \cite{DLM+:07}. The learning component of that earlier work was a brute-force exhaustive search over a concept class to find a hypothesis consistent with a sample of random examples. In this work, the learning component is a sophisticated exact learning algorithm for sparse $GF(2)$ polynomials due to Schapire and Sellie \cite{SchapireSellie:96}. A crucial element of this work, which enables us to simulate the membership queries required by \cite{SchapireSellie:96}, is an analysis establishing new properties of how sparse $GF(2)$ polynomials simplify under certain restrictions of low-influence'' sets of variables.

Postscript or pdf of long version of conference paper

pdf of journal version