Learning Unions of $\omega(1)$-Dimensional Rectangles
<|reference_start|>Learning Unions of $\omega(1)$-Dimensional Rectangles: We consider the problem of learning unions of rectangles over the domain $[b]^n$, in the uniform distribution membership query learning setting, where both b and n are "large". We obtain poly$(n, \log b)$-time algorithms for the following classes:   - poly$(n \log b)$-way Majority of $O(\frac{\log(n \log b)} {\log \log(n \log b)})$-dimensional rectangles.   - Union of poly$(\log(n \log b))$ many $O(\frac{\log^2 (n \log b)} {(\log \log(n \log b) \log \log \log (n \log b))^2})$-dimensional rectangles.   - poly$(n \log b)$-way Majority of poly$(n \log b)$-Or of disjoint $O(\frac{\log(n \log b)} {\log \log(n \log b)})$-dimensional rectangles.   Our main algorithmic tool is an extension of Jackson's boosting- and Fourier-based Harmonic Sieve algorithm [Jackson 1997] to the domain $[b]^n$, building on work of [Akavia, Goldwasser, Safra 2003]. Other ingredients used to obtain the results stated above are techniques from exact learning [Beimel, Kushilevitz 1998] and ideas from recent work on learning augmented $AC^{0}$ circuits [Jackson, Klivans, Servedio 2002] and on representing Boolean functions as thresholds of parities [Klivans, Servedio 2001].<|reference_end|>
@article{atici2005learning,
    title={Learning Unions of $\omega(1)$-Dimensional Rectangles},
    author={Alp Atici and Rocco A. Servedio},
    journal={Theoretical Computer Science, Vol. 405, No. 3, 209--222 (2008)},
    year={2005},
    doi={10.1016/j.tcs.2008.06.036},
    archivePrefix={arXiv},
    eprint={cs/0510038},
    primaryClass={cs.LG}
}