@inproceedings{
limbeck2024metric,
title={Metric Space Magnitude for Evaluating the Diversity of Latent Representations},
author={Katharina Limbeck and Rayna Andreeva and Rik Sarkar and Bastian Rieck},
booktitle={The Thirty-eighth Annual Conference on Neural Information Processing Systems},
year={2024},
url={https://openreview.net/forum?id=glgZZAfssH}
}
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
the automated estimation of diversity,
the detection of mode collapse, and
the evaluation of generative models for text, image, and graph data.
Metric Space Magnitude for Evaluating the Diversity of Latent Representations
Katharina Limbeck Rayna Andreeva Rik Sarkar Bastian Rieck