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.2022.53
URN: urn:nbn:de:0030-drops-156498
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2022/15649/
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De, Anindya ; Nadimpalli, Shivam ; Servedio, Rocco A.

Convex Influences

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LIPIcs-ITCS-2022-53.pdf (0.8 MB)

Abstract

We introduce a new notion of influence for symmetric convex sets over Gaussian space, which we term "convex influence". We show that this new notion of influence shares many of the familiar properties of influences of variables for monotone Boolean functions f: {±1}ⁿ → {±1}.
Our main results for convex influences give Gaussian space analogues of many important results on influences for monotone Boolean functions. These include (robust) characterizations of extremal functions, the Poincaré inequality, the Kahn-Kalai-Linial theorem [J. Kahn et al., 1988], a sharp threshold theorem of Kalai [G. Kalai, 2004], a stability version of the Kruskal-Katona theorem due to O'Donnell and Wimmer [R. O'Donnell and K. Wimmer, 2009], and some partial results towards a Gaussian space analogue of Friedgut’s junta theorem [E. Friedgut, 1998]. The proofs of our results for convex influences use very different techniques than the analogous proofs for Boolean influences over {±1}ⁿ. Taken as a whole, our results extend the emerging analogy between symmetric convex sets in Gaussian space and monotone Boolean functions from {±1}ⁿ to {±1}.

BibTeX - Entry

@InProceedings{de_et_al:LIPIcs.ITCS.2022.53,
  author =	{De, Anindya and Nadimpalli, Shivam and Servedio, Rocco A.},
  title =	{{Convex Influences}},
  booktitle =	{13th Innovations in Theoretical Computer Science Conference (ITCS 2022)},
  pages =	{53:1--53:21},
  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/2022/15649},
  URN =		{urn:nbn:de:0030-drops-156498},
  doi =		{10.4230/LIPIcs.ITCS.2022.53},
  annote =	{Keywords: Fourier analysis of Boolean functions, convex geometry, influences, threshold phenomena}
}

Keywords: Fourier analysis of Boolean functions, convex geometry, influences, threshold phenomena
Collection: 13th Innovations in Theoretical Computer Science Conference (ITCS 2022)
Issue Date: 2022
Date of publication: 25.01.2022

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