Metric Space Magnitude for Evaluating the Diversity of Latent Representations

TL;DR

This paper introduces a new metric space magnitude concept to quantify the diversity of latent representations, providing a stable, multi-scale measure that improves evaluation and detection tasks in generative modeling.

Contribution

It develops a novel magnitude-based framework for measuring diversity in latent spaces, formalizing a new dissimilarity notion and demonstrating its effectiveness across multiple data domains.

Findings

Superior performance in diversity estimation

Effective detection of mode collapse

Enhanced evaluation of generative models

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 (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.