Efficiently Testing Sparse GF(2) Polynomials
| dc.creator | Diakonikolas, Ilias | |
| dc.creator | Lee, Homin K. | |
| dc.creator | Matulef, Kevin | |
| dc.creator | Servedio, Rocco A. | |
| dc.creator | Wan, Andrew | |
| dc.date | 2008-05-13 | |
| dc.date.accessioned | 2026-07-07T09:38:33Z | |
| dc.date.available | 2026-07-07T09:38:33Z | |
| dc.description | We give the first algorithm that is both query-efficient and time-efficient for testing whether an unknown function $f: \{0,1\}^n \to \{0,1\}$ 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. | |
| dc.description | Full version of ICALP 2008 paper | |
| dc.identifier | https://arxiv.org/abs/0805.1765 | |
| dc.identifier | http://arxiv.org/abs/0805.1765 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/160844 | |
| dc.subject | Computational Complexity | |
| dc.title | Efficiently Testing Sparse GF(2) Polynomials | |
| dc.type | text |