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Simplicial Covering Dimension of Extremal Concept Classes

Authors Ari Blondal , Hamed Hatami , Pooya Hatami , Chavdar Lalov , Sivan Tretiak

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ACM Subject Classification
  • Computing methodologies → Supervised learning by classification
  • Theory of computation → Randomness, geometry and discrete structures
  • Mathematics of computing → Topology
Keywords
  • PAC Learning
  • Extremal Concept Classes
  • Replicability
  • List Replicability
  • Topology
  • Geometry

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Abstract

Dimension theory is a branch of topology concerned with defining and analyzing dimensions of geometric and topological spaces in purely topological terms. In this work, we adapt the classical notion of topological dimension (Lebesgue covering) to binary concept classes. The topological space naturally associated with a concept class is its space of realizable distributions. The loss function and the class itself induce a simplicial structure on this space, with respect to which we define a simplicial covering dimension.
We prove that for finite concept classes, this simplicial covering dimension exactly characterizes the list replicability number (equivalently, global stability) in PAC learning. This connection allows us to apply tools from classical dimension theory to compute the exact list replicability number of the broad family of extremal concept classes.

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Ari Blondal, Hamed Hatami, Pooya Hatami, Chavdar Lalov, and Sivan Tretiak. Simplicial Covering Dimension of Extremal Concept Classes. In 17th Innovations in Theoretical Computer Science Conference (ITCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 362, pp. 22:1-22:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.ITCS.2026.22

Author Details

Ari Blondal
  • McGill University, Montreal, Canada
Hamed Hatami
  • McGill University, Montreal, Canada
Pooya Hatami
  • Ohio State University, Columbus, OH, USA
Chavdar Lalov
  • Ohio State University, Columbus, OH, USA
Sivan Tretiak
  • Ohio State University, Columbus, OH, USA

Funding

  • Hatami, Hamed: Hamed Hatami is supported by an NSERC grant.

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