On Generation in Metric Spaces

2026-02-10 · Source: stat.ML updates on arXiv.org · Field: Technology & Digital — Artificial Intelligence & Machine Learning, Mathematics & Computational Sciences · Depth: Advanced, extended

Summary

This theoretical study extends the framework of generation in separable metric instance spaces, moving beyond prior models restricted to countable domains. It introduces the $(\varepsilon,\varepsilon^{\prime})$-closure dimension, a scale-sensitive analogue of closure dimension, to characterize uniform and non-uniform generatability and provide a sufficient condition for generation in the limit. The research reveals a sharp geometric contrast: in doubling spaces, including all finite-dimensional normed spaces, generatability is stable across novelty scales and invariant under equivalent metrics. However, in general metric spaces, such as the infinite-dimensional Hilbert space $\ell^{2}$, generatability can be highly scale-sensitive and metric-dependent, with all notions of generation potentially failing abruptly as novelty parameters vary. The work defines novelty through metric separation and allows asymmetric novelty parameters for the adversary and the generator, aligning with practical applications like protein structure generation where novelty is distance-based.

Key takeaway

For AI Researchers and Scientists developing generative models for continuous or high-dimensional data, recognize that the definition of "novelty" is crucial and context-dependent. Your choice of metric and novelty thresholds (adversary's $\varepsilon$ vs. generator's $\varepsilon^{\prime}$) significantly impacts model generatability, particularly outside of finite-dimensional Euclidean spaces. Be prepared for scale-sensitive and metric-dependent behaviors, and consider the implications for model robustness and generalization.

Key insights

Generatability in metric spaces is scale-sensitive and metric-dependent, especially in infinite-dimensional settings.

Principles

• Doubling spaces ensure stable generatability across scales and equivalent metrics.
• Asymmetric novelty parameters for adversary and generator are crucial in metric spaces.

Method

The $(\varepsilon,\varepsilon^{\prime})$-closure dimension characterizes generatability by measuring the largest number of $\varepsilon$-separated points whose closure has a finite $\varepsilon^{\prime}$-covering number.

In practice

• Generative models for 3D structures (e.g., proteins) require metric-space novelty definitions.
• Evaluating generative model generalization can use feature space distances for novelty.

Topics

• Metric Spaces
• Language Generation
• Generatability Theory
• Closure Dimension
• Doubling Spaces

Best for: AI Researcher, AI Scientist, Research Scientist

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Editorial summary, takeaway, and curation by AIssential. Original article published by stat.ML updates on arXiv.org.