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On Generation in Metric Spaces

Jiaxun Li, Vinod Raman, Ambuj Tewari

TL;DR

This work generalizes generation-in-the-limit from countable domains to separable metric spaces by introducing metric-based novelty via asymmetric parameters $\varepsilon$ and $\varepsilon'$. It defines the $C_{\varepsilon}^{\varepsilon'}(\mathcal{H})$ closure-dimension and shows that uniform generation in the limit is equivalent to having finite closure dimension, with non-uniform generation characterized through unions of finite-closure subclasses. A key finding is that in doubling spaces (e.g., finite-dimensional normed spaces) generation properties are stable across novelty scales and metrics, while in general metric spaces generation can be highly scale- and metric-dependent; explicit constructions in $\ell^2$ illustrate abrupt regime changes. These results formalize how geometric structure shapes what can be generated and under what conditions, bridging theoretical insights with potential applications to continuous-domain generation tasks. The work also discusses algorithmic extensions and outlines directions for handling symmetries and broader geometric conditions beyond doubling.

Abstract

We study generation in separable metric instance spaces. We extend the language generation framework from Kleinberg and Mullainathan [2024] beyond countable domains by defining novelty through metric separation and allowing asymmetric novelty parameters for the adversary and the generator. We introduce the $(\varepsilon,\varepsilon')$-closure dimension, a scale-sensitive analogue of closure dimension, which yields characterizations of uniform and non-uniform generatability and a sufficient condition for generation in the limit. Along the way, we identify a sharp geometric contrast. Namely, in doubling spaces, including all finite-dimensional normed spaces, generatability is stable across novelty scales and invariant under equivalent metrics. In general metric spaces, however, generatability can be highly scale-sensitive and metric-dependent; even in the natural infinite-dimensional Hilbert space $\ell^2$, all notions of generation may fail abruptly as the novelty parameters vary.

On Generation in Metric Spaces

TL;DR

This work generalizes generation-in-the-limit from countable domains to separable metric spaces by introducing metric-based novelty via asymmetric parameters and . It defines the closure-dimension and shows that uniform generation in the limit is equivalent to having finite closure dimension, with non-uniform generation characterized through unions of finite-closure subclasses. A key finding is that in doubling spaces (e.g., finite-dimensional normed spaces) generation properties are stable across novelty scales and metrics, while in general metric spaces generation can be highly scale- and metric-dependent; explicit constructions in illustrate abrupt regime changes. These results formalize how geometric structure shapes what can be generated and under what conditions, bridging theoretical insights with potential applications to continuous-domain generation tasks. The work also discusses algorithmic extensions and outlines directions for handling symmetries and broader geometric conditions beyond doubling.

Abstract

We study generation in separable metric instance spaces. We extend the language generation framework from Kleinberg and Mullainathan [2024] beyond countable domains by defining novelty through metric separation and allowing asymmetric novelty parameters for the adversary and the generator. We introduce the -closure dimension, a scale-sensitive analogue of closure dimension, which yields characterizations of uniform and non-uniform generatability and a sufficient condition for generation in the limit. Along the way, we identify a sharp geometric contrast. Namely, in doubling spaces, including all finite-dimensional normed spaces, generatability is stable across novelty scales and invariant under equivalent metrics. In general metric spaces, however, generatability can be highly scale-sensitive and metric-dependent; even in the natural infinite-dimensional Hilbert space , all notions of generation may fail abruptly as the novelty parameters vary.
Paper Structure (40 sections, 26 theorems, 132 equations, 1 figure, 1 algorithm)

This paper contains 40 sections, 26 theorems, 132 equations, 1 figure, 1 algorithm.

Table of Contents

  1. Introduction
  2. Main Contributions
  3. Related Work
  4. Uniform and Non-Uniform Generation
  5. Extensions of Language Generation
  6. Other Related Work
  7. Preliminaries
  8. Example Space and Hypothesis Class
  9. Generatability
  10. Characterizations of Generatability
  11. Properties of Generatability
  12. Generatability is Well Behaved in Doubling Spaces
  13. Generatability is Scale-Sensitive
  14. Generatability is Metric-Dependent
  15. Discussion
  16. ...and 25 more sections

Key Result

Lemma 2.1

Let $\mathcal{H}\subseteq \{0,1\}^\mathcal{X}$ be any class satisfying the $r$-UUS property for an $r>0$. Then for any positive $\delta<r$, $\mathcal{H}$ satisfies the $\delta$-UUS property.

Figures (1)

  • Figure 1: Illustration of two generation regimes in a metric space. The horizontal axis represents the adversary parameter, and the vertical axis represents the generator parameter.

Theorems & Definitions (72)

  • Definition 2.1: Closed Ball
  • Definition 2.2: Covering Number
  • Definition 2.3: $r$-Uniformly Unbounded Support ($r$-UUS)
  • Lemma 2.1
  • Definition 2.4: Generator
  • Definition 2.5: Generatability in the Limit
  • Remark 2.1
  • Definition 2.6: Uniform Generatability
  • Definition 2.7: Non-Uniform Generatability
  • Definition 3.1: $(\varepsilon,\varepsilon')$-Closure dimension
  • ...and 62 more