The magnitude of a metric space is a recently-established invariant, providing a measure of the 'effective size' of a space across multiple scales while also capturing numerous geometrical properties. We develop a family of magnitude-based measures of the intrinsic diversity of latent representations, formalising a novel notion of dissimilarity between magnitude functions of finite metric spaces. Our measures are provably stable under perturbations of the data, can be efficiently calculated, and enable a rigorous multi-scale comparison of latent representations. We show the utility and superior performance of our measures in an experimental suite that comprises different domains and tasks, including the evaluation of diversity, the detection of mode collapse, and the evaluation of generative models for text, image, and graph data.

翻译：度量空间的量级是近期确立的一个不变量，能够在多个尺度上衡量空间的“有效大小”，同时捕捉多种几何性质。我们基于量级开发了一套衡量潜在表示内在多样性的方法，将有限度量空间量级函数之间的一种新异质性概念形式化。我们的度量方法在数据扰动下具有可证明的稳定性，能够高效计算，并支持对潜在表示进行严格的多尺度比较。通过涵盖不同领域和任务的实验套件——包括多样性评估、模式坍塌检测以及文本、图像和图数据生成模型的评估——我们展示了该方法的实用性和优越性能。