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.

翻译：本文研究可分离度量实例空间中的生成问题。我们将Kleinberg和Mullainathan[2024]提出的语言生成框架从可数域推广到更一般的度量空间：通过度量分离定义新颖性概念，并允许对抗方与生成方采用非对称的新颖性参数。我们引入了$(\varepsilon,\varepsilon')$-闭包维数——这是闭包维数的一种尺度敏感型推广，该概念可导出均匀生成与非均匀生成的特征刻画，并为极限情形下的生成提供充分条件。研究过程中，我们揭示了一个鲜明的几何对比现象：在加倍空间（包括所有有限维赋范空间）中，生成性在不同新颖性尺度下保持稳定，且在等价度量下具有不变性；然而在一般度量空间中，生成性可能具有强烈的尺度敏感性且依赖于具体度量——即使在自然的无限维希尔伯特空间$\ell^2$中，所有生成概念都可能随着新颖性参数的微小变化而突然失效。