Metric Space Magnitude for Evaluating the Diversity of Latent Representations

Katharina Limbeck, Rayna Andreeva, Rik Sarkar, Bastian Rieck

Keywords: diversity evaluation, generative model evaluation, metric space magnitude, geometric machine learning

TL;DR: We develop novel diversity measures for evaluating latent representations based on metric space magnitude, a novel geometric invariant.

Abstract:

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.

Primary Area: Evaluation (methodology, meta studies, replicability and validity)

Submission Number: 16881