From Topology to Retrieval: Decoding Embedding Spaces with Unified Signatures

2025-11-27

AI Summary

This work addresses the weak interpretability of text embedding spaces and their limited structural representation. We propose the Unified Topological Signature (UTS) framework—the first systematic approach to jointly model the topological and geometric structure of embedding spaces. UTS integrates multi-dimensional features, including persistent homology, curvature estimation, and local density, overcoming the redundancy and low discriminability of conventional metrics. By applying clustering analysis and correlation modeling, UTS decodes the mapping between spatial organization and downstream retrieval performance, establishing a quantitative relationship between topological features and document retrievability. Extensive evaluation across multiple state-of-the-art embedding models and benchmark datasets demonstrates that UTS stably predicts inter-model performance differences and ranking effectiveness, exhibiting strong generalization capability and cross-model comparability.

Technology Category

• Machine Learning: Representation Learning
• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning
• Natural Language Processing: Sentence-level Semantics, Textual Inference, etc.

Application Category

• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Search and Retrieval-Augmented AI: Web query analysis, representation and understanding
• Web Mining and Content Analysis: Bridging structured and unstructured data

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.

Problem

• Analyzing topological and geometric measures of text embedding spaces
• Introducing a unified framework to characterize embedding spaces holistically
• Linking topological structure to retrieval performance and model properties

Innovation

• Unified Topological Signatures characterize embedding spaces holistically
• Predicts model properties and architectural similarities effectively
• Links topological structure to retrieval performance and document retrievability