We construct and analyze a new pseudorandom generator for degree 2 polynomial threshold functions with respect to the Gaussian measure. In particular, we obtain one whose seed length is polylogarithmic in both the dimension and the desired error, a substantial improvement over existing constructions. Our generator is obtained as an appropriate weighted average of pseudorandom generators against read once branching programs. The analysis requires a number of ideas including a hybrid argument and a structural result that allows us to treat our degree 2 threshold function as a function of a number of linear polynomials and one approximately linear polynomial.
@InProceedings{kane:LIPIcs.CCC.2015.567, author = {Kane, Daniel M.}, title = {{A Polylogarithmic PRG for Degree 2 Threshold Functions in the Gaussian Setting}}, booktitle = {30th Conference on Computational Complexity (CCC 2015)}, pages = {567--581}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-81-1}, ISSN = {1868-8969}, year = {2015}, volume = {33}, editor = {Zuckerman, David}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2015.567}, URN = {urn:nbn:de:0030-drops-50534}, doi = {10.4230/LIPIcs.CCC.2015.567}, annote = {Keywords: polynomial threshold function, pseudorandom generator, Gaussian distribution} }
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