On Generation in Metric Spaces

2026-02-07

AI Summary

This study investigates the feasibility and stability of generation problems in separable metric spaces, with a particular focus on characterizing generative capacity over uncountable domains. By introducing a novel notion of metric separation–based novelty and an asymmetric novelty parameter, the authors propose the $(\varepsilon, \varepsilon')$-closure dimension to unify the description of both uniform and non-uniform generatability. Through theoretical analysis that integrates properties of metric spaces, closure dimensions, doubling spaces, and Hilbert space geometry, the work reveals a fundamental disparity in the sensitivity of generative stability to scale and metric perturbations: in doubling spaces—such as finite-dimensional normed spaces—generative capacity exhibits robustness to scaling, whereas in infinite-dimensional spaces like $\ell^2$, it may collapse under arbitrarily small parameter variations.

Technology Category

• Machine Learning: Deep Generative Models & Autoencoders
• Natural Language Processing: Generation
• Computer Vision: Generative Adversarial Networks (GANs) for Vision

Application Category

• Economics, Online Markets and Human Computation: Economic ramifications for generative AI infrastructure and applications
• Security and Privacy: Large-scale security measurements
• Social Networks and Social Media: Generative AI / large language models and their impact on social systems

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.

Problem

• generation
• metric spaces
• novelty
• scale sensitivity
• closure dimension

Innovation

• metric generation
• closure dimension
• novelty parameter
• doubling space
• scale sensitivity