License: Creative Commons Attribution 3.0 Unported license (CC BY 3.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.CCC.2018.28
URN: urn:nbn:de:0030-drops-88616
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2018/8861/
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Ghazi, Badih ; Kamath, Pritish ; Raghavendra, Prasad

Dimension Reduction for Polynomials over Gaussian Space and Applications

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Abstract

We introduce a new technique for reducing the dimension of the ambient space of low-degree polynomials in the Gaussian space while preserving their relative correlation structure. As an application, we obtain an explicit upper bound on the dimension of an epsilon-optimal noise-stable Gaussian partition. In fact, we address the more general problem of upper bounding the number of samples needed to epsilon-approximate any joint distribution that can be non-interactively simulated from a correlated Gaussian source. Our results significantly improve (from Ackermann-like to "merely" exponential) the upper bounds recently proved on the above problems by De, Mossel & Neeman [CCC 2017, SODA 2018 resp.] and imply decidability of the larger alphabet case of the gap non-interactive simulation problem posed by Ghazi, Kamath & Sudan [FOCS 2016].
Our technique of dimension reduction for low-degree polynomials is simple and can be seen as a generalization of the Johnson-Lindenstrauss lemma and could be of independent interest.

BibTeX - Entry

@InProceedings{ghazi_et_al:LIPIcs:2018:8861,
  author =	{Badih Ghazi and Pritish Kamath and Prasad Raghavendra},
  title =	{{Dimension Reduction for Polynomials over Gaussian Space and Applications}},
  booktitle =	{33rd Computational Complexity Conference (CCC 2018)},
  pages =	{28:1--28:37},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-069-9},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{102},
  editor =	{Rocco A. Servedio},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/opus/volltexte/2018/8861},
  URN =		{urn:nbn:de:0030-drops-88616},
  doi =		{10.4230/LIPIcs.CCC.2018.28},
  annote =	{Keywords: Dimension reduction, Low-degree Polynomials, Noise Stability, Non-Interactive Simulation}
}

Keywords: Dimension reduction, Low-degree Polynomials, Noise Stability, Non-Interactive Simulation
Collection: 33rd Computational Complexity Conference (CCC 2018)
Issue Date: 2018
Date of publication: 04.06.2018

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