On Generation in Metric Spaces

Jiaxun Li; Vinod Raman; Ambuj Tewari

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arXiv · 2602.07710

On Generation in Metric Spaces

Jiaxun Li·Vinod Raman·Ambuj Tewari

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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.

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Jiaxun Li, Vinod Raman, Ambuj Tewari. 2026-02-07. On Generation in Metric Spaces. https://arxiv.org/abs/2602.07710

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