The magnitude of a metric space is a novelinvariant that provides a measure of the'effective size' of a space acrossmultiple scales, while also capturing numerous geometrical properties, such as curvature, density, or entropy.We develop a family of magnitude-based measures of the intrinsicdiversity of latent representations, formalising a novel notion ofdissimilarity between magnitude functions of finite metric spaces.Our measures are provably stable under perturbations of the data, can beefficiently calculated, and enable a rigorous multi-scale characterisation and comparison oflatent representations. We show their utility and superior performance across different domains and tasks, includingthe automated estimation of diversity,the detection of mode collapse, andthe evaluation of generative models for text, image, and graph data.

Metric space magnitude, an active subject of research in algebraic topology, originally arose in the context of biology, where it was used to represent the effective number of distinct species in an environment. In a more general setting, the magnitude of a metric space is a real number that aims to quantify the effective number of distinct points in the space. The contribution of each point to a metric space's global magnitude, which is encoded by the {\em weighting vector}, captures much of the underlying geometry of the original metric space. Surprisingly, when the metric space is Euclidean, the weighting vector also serves as an effective tool for boundary detection. This allows the weighting vector to serve as the foundation of novel algorithms for classic machine learning tasks such as classification, outlier detection and active learning. We demonstrate, using experiments and comparisons on classic benchmark datasets, the promise of the proposed magnitude and weighting vector-based approaches.