Metric Space Magnitude for Evaluating the Diversity of Latent Representations

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.