From Topology to Retrieval: Decoding Embedding Spaces with Unified Signatures

TL;DR

This paper introduces Unified Topological Signatures (UTS), a comprehensive framework for analyzing and characterizing the structure of text embedding spaces, improving interpretability and retrieval performance understanding.

Contribution

The paper proposes UTS, a novel holistic framework that captures the topological structure of embedding spaces and predicts model properties and retrieval effectiveness.

Findings

High redundancy among topological measures.

Single metrics often insufficient to differentiate embeddings.

UTS effectively predicts retrievability and model similarities.

Abstract

Studying how embeddings are organized in space not only enhances model interpretability but also uncovers factors that drive downstream task performance. In this paper, we present a comprehensive analysis of topological and geometric measures across a wide set of text embedding models and datasets. We find a high degree of redundancy among these measures and observe that individual metrics often fail to sufficiently differentiate embedding spaces. Building on these insights, we introduce Unified Topological Signatures (UTS), a holistic framework for characterizing embedding spaces. We show that UTS can predict model-specific properties and reveal similarities driven by model architecture. Further, we demonstrate the utility of our method by linking topological structure to ranking effectiveness and accurately predicting document retrievability. We find that a holistic, multi-attribute perspective is essential to understanding and leveraging the geometry of text embeddings.