On Generation in Metric Spaces

• URL: http://arxiv.org/abs/2602.07710v1
• Date: Sat, 07 Feb 2026 21:37:59 GMT
• Title: On Generation in Metric Spaces
• Authors: Jiaxun Li, Vinod Raman, Ambuj Tewari,
• Abstract summary: We study generation in separable metric instance spaces.<n>We introduce the $(varepsilon,varepsilon')$-closure dimension.<n>In general metric spaces, generatability can be highly scale-sensitive and metric-dependent.
• Score: 29.71858964413957
• License: http://creativecommons.org/licenses/by/4.0/
• 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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