The magnitude of a metric space is a novel invariant that provides a measure of the 'effective size' of a space across multiple scales, while also capturing numerous geometrical properties, such as curvature, density, or entropy. 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 characterisation and comparison of latent representations. We show their utility and superior performance across different domains and tasks, including (i) the automated estimation of diversity, (ii) the detection of mode collapse, and (iii) the evaluation of generative models for text, image, and graph data.

翻译：度量空间幅度是一种新颖的不变量，它提供了空间在多个尺度上的“有效尺寸”度量，同时捕捉了众多几何特性，如曲率、密度或熵。我们开发了一系列基于幅度的度量，用于衡量潜在表示的内在多样性，形式化地定义了有限度量空间幅度函数之间差异性的新概念。我们的度量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表征与比较。我们展示了这些度量在不同领域和任务中的实用性与优越性能，包括：（i）多样性的自动估计，（ii）模式崩溃的检测，以及（iii）针对文本、图像和图数据的生成模型评估。