On Generation in Metric Spaces

arXiv:2602.07710v1 Announce Type: new
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.