Simplicial covering dimension of extremal concept classes

2025-11-14

AI Summary

This paper precisely characterizes the list-reproducibility number (i.e., global stability) of binary concept classes in PAC learning. To this end, it introduces the *simplicial covering dimension*—a novel topological dimension obtained by generalizing the classical Lebesgue covering dimension to the simplicial complex induced by a concept class. This dimension depends solely on the combinatorial structure of the class and is distribution-independent. It is rigorously proven to equal the minimal list-reproducibility number for finite concept classes. Methodologically, the work integrates topological dimension theory with combinatorial analysis, leveraging the geometric structure of the realizable distribution space and the loss-induced simplicial complex. The core contribution is a deep correspondence between computational learning theory and classical dimension theory: it yields the first exact formula for the list-reproducibility number of extremal concept classes and provides a computable, intrinsic criterion for global stability.

Technology Category

• Machine Learning: Learning Theory
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Constraint Learning and Acquisition

Application Category

• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

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.

Problem

• Adapting topological dimension theory to binary concept classes
• Defining simplicial covering dimension for concept class spaces
• Characterizing list replicability in PAC learning using dimension theory

Innovation

• Adapted topological dimension to concept classes
• Defined simplicial covering dimension for distributions
• Connected dimension theory to PAC learning replicability