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Variants of VC dimension and their applications to dynamics

Guorong Gao, Jie Ma, Mingyuan Rong, Tuan Tran

TL;DR

This paper studies two VC-dimension variants—the $k$-Natarajan dimension and a partial-concept-classes VC variant—and develops general Sauer–Shelah-type bounds that unify several extremal results in learning theory. It proves a new bound for $\dim_k(\mathcal{H})$ that extends the classical VC, Natarajan, and Steele bounds and uses it to strengthen a central dynamical-systems theorem of Huang and Ye. It also provides a concise proof of a key lemma for partial concept classes and shows a polynomial-in-$n$ bound on covering numbers $C(\mathcal{H})$, enabling applications to topological and symbolic dynamics via maximal pattern entropy $h^*_{top}(T)$. The work draws connections among combinatorics, dynamical systems, and communication complexity, offering a unified framework for analyzing pattern complexity across these domains $-$ with implications for entropy theory and learning-theoretic bounds.

Abstract

Since its introduction by Vapnik and Chervonenkis in the 1960s, the VC dimension and its variants have played a central role in numerous fields. In this paper, we investigate several variants of the VC dimension and their applications to dynamical systems. First, we prove a new bound for a recently introduced generalization of VC dimension, which unifies and extends various extremal results on the VC, Natarajan, and Steele dimensions. This new bound allows us to strengthen one of the main theorems of Huang and Ye [Adv. Math., 2009] in dynamical systems. Second, we refine a key lemma of Huang and Ye related to a variant of VC dimension by providing a more concise and conceptual proof. We also highlight a surprising connection among this result, combinatorics, dynamical systems, and recent advances in communication complexity.

Variants of VC dimension and their applications to dynamics

TL;DR

This paper studies two VC-dimension variants—the -Natarajan dimension and a partial-concept-classes VC variant—and develops general Sauer–Shelah-type bounds that unify several extremal results in learning theory. It proves a new bound for that extends the classical VC, Natarajan, and Steele bounds and uses it to strengthen a central dynamical-systems theorem of Huang and Ye. It also provides a concise proof of a key lemma for partial concept classes and shows a polynomial-in- bound on covering numbers , enabling applications to topological and symbolic dynamics via maximal pattern entropy . The work draws connections among combinatorics, dynamical systems, and communication complexity, offering a unified framework for analyzing pattern complexity across these domains with implications for entropy theory and learning-theoretic bounds.

Abstract

Since its introduction by Vapnik and Chervonenkis in the 1960s, the VC dimension and its variants have played a central role in numerous fields. In this paper, we investigate several variants of the VC dimension and their applications to dynamical systems. First, we prove a new bound for a recently introduced generalization of VC dimension, which unifies and extends various extremal results on the VC, Natarajan, and Steele dimensions. This new bound allows us to strengthen one of the main theorems of Huang and Ye [Adv. Math., 2009] in dynamical systems. Second, we refine a key lemma of Huang and Ye related to a variant of VC dimension by providing a more concise and conceptual proof. We also highlight a surprising connection among this result, combinatorics, dynamical systems, and recent advances in communication complexity.
Paper Structure (14 sections, 16 theorems, 41 equations)

This paper contains 14 sections, 16 theorems, 41 equations.

Table of Contents

  1. Introduction
  2. A generalization of VC dimension: k-Natarajan dimension
  3. A variant of VC dimension to partial concept classes
  4. Applications to dynamical systems
  5. Notation and organization
  6. -Natarajan dimension
  7. Construction
  8. A variant of VC dimension to partial concept classes
  9. Applications to dynamical systems
  10. An application in topological dynamical systems
  11. An application in symbolic dynamics
  12. Concluding remarks
  13. Proof of
  14. Proof of

Key Result

Theorem 1.1

Let $r\ge k \ge 2$ and $n\ge d\ge 1$. For any family $\mathcal{H}\subseteq \{1,\ldots,r\}^{[n]}$ with $\dim_k(\mathcal{H}) \le d$,

Theorems & Definitions (30)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3: AHHM21
  • Conjecture 1.4: HY09
  • Theorem 1.5: HY09
  • Theorem 1.6
  • Theorem 1.7: HY09
  • Lemma 2.1
  • proof : Proof of \ref{['thm:Natarajan']} assuming \ref{['lem:branching']}
  • proof : Proof of \ref{['lem:branching']} (continuation)
  • ...and 20 more