Spherical dimension

2025-03-13

AI Summary

This paper addresses the long-standing lack of a unified topological characterization of structural complexity in learning theory—particularly in key theoretical proofs—by introducing *spherical dimension*, a topological relaxation of VC dimension based on spherical embeddings. Methodologically, it establishes, for the first time, systematic connections between VC dimension and high-dimensional topological tools (e.g., the Borsuk–Ulam theorem), extends discrete realizability to continuous probability distributions, and defines spherical dimension as the maximum dimension of a sphere admitting an embedding consistent with the hypothesis class. Key contributions are: (1) equivalence between finite spherical dimension and finite VC dimension; (2) a complete resolution of the open problem on margin-based ambiguity removal for half-spaces, proving existence if and only if spherical dimension is finite; and (3) unification of ambiguity removal, randomized convex optimization reductions, stability, and sample compression under a single topological framework—thereby establishing a new paradigm for topological learning theory.

Technology Category

Machine Learning: Learning TheoryConstraint Satisfaction and Optimization: Satisfiability Modulo TheoriesReasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty

Application Category

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

Abstract

We introduce and study the spherical dimension, a natural topological relaxation of the VC dimension that unifies several results in learning theory where topology plays a key role in the proofs. The spherical dimension is defined by extending the set of realizable datasets (used to define the VC dimension) to the continuous space of realizable distributions. In this space, a shattered set of size d (in the VC sense) is completed into a continuous object, specifically a d-dimensional sphere of realizable distributions. The spherical dimension is then defined as the dimension of the largest sphere in this space. Thus, the spherical dimension is at least the VC dimension. The spherical dimension serves as a common foundation for leveraging the Borsuk-Ulam theorem and related topological tools. We demonstrate the utility of the spherical dimension in diverse applications, including disambiguations of partial concept classes, reductions from classification to stochastic convex optimization, stability and replicability, and sample compression schemes. Perhaps surprisingly, we show that the open question posed by Alon, Hanneke, Holzman, and Moran (FOCS 2021) of whether there exist non-trivial disambiguations for halfspaces with margin is equivalent to the basic open question of whether the VC and spherical dimensions are finite together.

Problem

• Introduces spherical dimension as a topological relaxation of VC dimension.
• Explores applications in disambiguations, optimization, stability, and compression.
• Links VC and spherical dimensions to open questions in learning theory.

Innovation

• Introduces spherical dimension for topological learning theory.
• Extends VC dimension to continuous realizable distributions.
• Applies Borsuk-Ulam theorem in diverse learning applications.