Topological Metric for Unsupervised Embedding Quality Evaluation

TL;DR

This paper introduces Persistence, a topology-based metric using persistent homology to evaluate the quality of unsupervised embeddings by capturing their geometric and topological properties, correlating well with downstream tasks.

Contribution

It presents a novel, topology-aware metric for unsupervised embedding evaluation that outperforms existing methods in correlating with downstream performance.

Findings

Persistence achieves top-tier correlation with downstream task performance.

It outperforms existing unsupervised metrics in diverse domains.

Enables reliable model and hyperparameter selection without labels.

Abstract

Modern representation learning increasingly relies on unsupervised and self-supervised methods trained on large-scale unlabeled data. While these approaches achieve impressive generalization across tasks and domains, evaluating embedding quality without labels remains an open challenge. In this work, we propose Persistence, a topology-aware metric based on persistent homology that quantifies the geometric structure and topological richness of embedding spaces in a fully unsupervised manner. Unlike metrics that assume linear separability or rely on covariance structure, Persistence captures global and multi-scale organization. Empirical results across diverse domains show that Persistence consistently achieves top-tier correlations with downstream performance, outperforming existing unsupervised metrics and enabling reliable model and hyperparameter selection.