License: Creative Commons Attribution 4.0 International license (CC BY 4.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.ITCS.2022.89
URN: urn:nbn:de:0030-drops-156853
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2022/15685/
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Jones, Chris ; Potechin, Aaron

Almost-Orthogonal Bases for Inner Product Polynomials

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LIPIcs-ITCS-2022-89.pdf (0.7 MB)

Abstract

In this paper, we consider low-degree polynomials of inner products between a collection of random vectors. We give an almost orthogonal basis for this vector space of polynomials when the random vectors are Gaussian, spherical, or Boolean. In all three cases, our basis admits an interesting combinatorial description based on the topology of the underlying graph of inner products.
We also analyze the expected value of the product of two polynomials in our basis. In all three cases, we show that this expected value can be expressed in terms of collections of matchings on the underlying graph of inner products. In the Gaussian and Boolean cases, we show that this expected value is always non-negative. In the spherical case, we show that this expected value can be negative but we conjecture that if the underlying graph of inner products is planar then this expected value will always be non-negative.

BibTeX - Entry

@InProceedings{jones_et_al:LIPIcs.ITCS.2022.89,
  author =	{Jones, Chris and Potechin, Aaron},
  title =	{{Almost-Orthogonal Bases for Inner Product Polynomials}},
  booktitle =	{13th Innovations in Theoretical Computer Science Conference (ITCS 2022)},
  pages =	{89:1--89:21},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-217-4},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{215},
  editor =	{Braverman, Mark},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/opus/volltexte/2022/15685},
  URN =		{urn:nbn:de:0030-drops-156853},
  doi =		{10.4230/LIPIcs.ITCS.2022.89},
  annote =	{Keywords: Orthogonal polynomials, Fourier analysis, combinatorics}
}

Keywords: Orthogonal polynomials, Fourier analysis, combinatorics
Collection: 13th Innovations in Theoretical Computer Science Conference (ITCS 2022)
Issue Date: 2022
Date of publication: 25.01.2022

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