Towards Robust Trajectory Embedding for Similarity Computation: When Triangle Inequality Violations in Distance Metrics Matter

2025-04-15

AI Summary

Traditional Euclidean-space trajectory similarity computation is constrained by the triangle inequality, resulting in poor embedding robustness and limited performance in trajectory retrieval and clustering. To address this, we introduce hyperbolic geometry—previously unexplored for trajectory representation learning—leveraging its inherent capacity to model hierarchical and sparse trajectory structures. We propose a Lorentzian-distance-based hyperbolic trajectory embedding method; design an agnostic, plug-and-play LH-plugin framework; and incorporate cosh-optimized projection with a dynamic fusion distance mechanism to adaptively mitigate triangle inequality distortion. Evaluated on multiple real-world trajectory datasets, our approach consistently enhances similarity computation accuracy across mainstream models, significantly improving robustness in trajectory retrieval and clustering. This work establishes a scalable, high-precision geometric modeling paradigm for trajectory analysis.

Technology Category

• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning
• Machine Learning: Learning with Manifolds
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems

Application Category

• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs
• Search and Retrieval-Augmented AI: Web query analysis, representation and understanding

Abstract

Trajectory similarity is a cornerstone of trajectory data management and analysis. Traditional similarity functions often suffer from high computational complexity and a reliance on specific distance metrics, prompting a shift towards deep representation learning in Euclidean space. However, existing Euclidean-based trajectory embeddings often face challenges due to the triangle inequality constraints that do not universally hold for trajectory data. To address this issue, this paper introduces a novel approach by incorporating non-Euclidean geometry, specifically hyperbolic space, into trajectory representation learning. We present the first-ever integration of hyperbolic space to resolve the inherent limitations of the triangle inequality in Euclidean embeddings. In particular, we achieve it by designing a Lorentz distance measure, which is proven to overcome triangle inequality constraints. Additionally, we design a model-agnostic framework LH-plugin to seamlessly integrate hyperbolic embeddings into existing representation learning pipelines. This includes a novel projection method optimized with the Cosh function to prevent the diminishment of distances, supported by a theoretical foundation. Furthermore, we propose a dynamic fusion distance that intelligently adapts to variations in triangle inequality constraints across different trajectory pairs, blending Lorentzian and Euclidean distances for more robust similarity calculations. Comprehensive experimental evaluations demonstrate that our approach effectively enhances the accuracy of trajectory similarity measures in state-of-the-art models across multiple real-world datasets. The LH-plugin not only addresses the triangle inequality issues but also significantly refines the precision of trajectory similarity computations, marking a substantial advancement in the field of trajectory representation learning.

Problem

• Overcoming triangle inequality violations in trajectory similarity metrics
• Reducing computational complexity in trajectory data analysis
• Integrating non-Euclidean geometry for robust trajectory embeddings

Innovation

• Uses hyperbolic space for trajectory embedding
• Introduces Lorentz distance to overcome constraints
• Dynamic fusion distance adapts to variations