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Dimension Reduction for Polynomials over Gaussian Space and Applications

Authors Badih Ghazi, Pritish Kamath, Prasad Raghavendra

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Subject Classification

ACM Subject Classification
  • Theory of computation → Complexity theory and logic
Keywords
  • Dimension reduction
  • Low-degree Polynomials
  • Noise Stability
  • Non-Interactive Simulation

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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.

Cite As Get BibTex

Badih Ghazi, Pritish Kamath, and Prasad Raghavendra. Dimension Reduction for Polynomials over Gaussian Space and Applications. In 33rd Computational Complexity Conference (CCC 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 102, pp. 28:1-28:37, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018) https://doi.org/10.4230/LIPIcs.CCC.2018.28

Author Details

Badih Ghazi
  • Google Research, 1600 Amphitheatre Parkway Mountain View, CA 94043, USA
Pritish Kamath
  • Massachusetts Institute of Technology, 77 Massachusetts Ave, Cambridge, MA 02139, USA
Prasad Raghavendra
  • University of California Berkeley, Berkeley, CA, USA

Funding

  • Ghazi, Badih: The work was done while the author was a student at MIT. Supported in parts by NSF CCF-1650733 and CCF-1420692.
  • Kamath, Pritish: Supported in parts by NSF CCF-1420956, CCF-1420692, CCF-1218547 and CCF-1650733.
  • Raghavendra, Prasad: Research supported by Okawa Research Grant and NSF CCF-1408643.

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