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.ICALP.2019.34
URN: urn:nbn:de:0030-drops-106105
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2019/10610/
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Chalopin, Jérémie ; Chepoi, Victor ; Moran, Shay ; Warmuth, Manfred K.

Unlabeled Sample Compression Schemes and Corner Peelings for Ample and Maximum Classes

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Abstract

We examine connections between combinatorial notions that arise in machine learning and topological notions in cubical/simplicial geometry. These connections enable to export results from geometry to machine learning. Our first main result is based on a geometric construction by H. Tracy Hall (2004) of a partial shelling of the cross-polytope which can not be extended. We use it to derive a maximum class of VC dimension 3 that has no corners. This refutes several previous works in machine learning from the past 11 years. In particular, it implies that the previous constructions of optimal unlabeled compression schemes for maximum classes are erroneous.
On the positive side we present a new construction of an optimal unlabeled compression scheme for maximum classes. We leave as open whether our unlabeled compression scheme extends to ample (a.k.a. lopsided or extremal) classes, which represent a natural and far-reaching generalization of maximum classes. Towards resolving this question, we provide a geometric characterization in terms of unique sink orientations of the 1-skeletons of associated cubical complexes.

BibTeX - Entry

@InProceedings{chalopin_et_al:LIPIcs:2019:10610,
  author =	{J{\'e}r{\'e}mie Chalopin and Victor Chepoi and Shay Moran and Manfred K. Warmuth},
  title =	{{Unlabeled Sample Compression Schemes and Corner Peelings for Ample and Maximum Classes}},
  booktitle =	{46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)},
  pages =	{34:1--34:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-109-2},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{132},
  editor =	{Christel Baier and Ioannis Chatzigiannakis and Paola Flocchini and Stefano Leonardi},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2019/10610},
  URN =		{urn:nbn:de:0030-drops-106105},
  doi =		{10.4230/LIPIcs.ICALP.2019.34},
  annote =	{Keywords: VC-dimension, sample compression, Sauer-Shelah-Perles lemma, Sandwich lemma, maximum class, ample/extremal class, corner peeling, unique sink orienta}
}

Keywords: VC-dimension, sample compression, Sauer-Shelah-Perles lemma, Sandwich lemma, maximum class, ample/extremal class, corner peeling, unique sink orienta
Collection: 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)
Issue Date: 2019
Date of publication: 04.07.2019

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