研究度量空间中的生成机制，揭示了新颖性尺度对生成能力的决定性影响。
On Generation in Metric Spaces
- 用度量距离定义新颖性，支持生成器与对手的不对称参数设置
- 提出(ε,ε')-闭包维数，可刻画均匀与非均匀可生成性
- 发现高维空间中生成能力对尺度和度量变化极度敏感
我们研究可分度量空间中的生成问题。通过将Kleinberg和Mullainathan [2024]的语言生成框架从可数域扩展到一般度量空间，引入基于度量分离的新颖性定义，并允许生成器与对手采用不对称的新颖性参数。提出(ε,ε')-闭包维数这一尺度敏感的闭包维数类比，实现了对均匀与非均匀可生成性的表征，并给出极限生成的充分条件。研究发现显著的几何差异：在双倍空间（包括所有有限维范数空间）中，生成能力在不同新颖性尺度下保持稳定，且对等价度量不变；而在一般度量空间中，生成能力可能高度依赖尺度与度量选择；甚至在自然的无限维希尔伯特空间ℓ²中，各类生成概念可能随新颖性参数变化而突然失效。
原文摘要 · Abstract (English)
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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