Tracking Temporal Dynamics of Vector Sets with Gaussian Process

2025-12-17

AI Summary

This paper addresses the challenge of dynamically modeling time-varying vector sets—such as spatiotemporal crime distributions or evolving word embeddings. We propose a distribution-level temporal modeling framework based on infinite-dimensional Gaussian processes (GPs). Unlike conventional point-wise approaches, our method treats each time-point’s vector set as a probability distribution and employs kernel embedding with random Fourier features to achieve scalable, low-dimensional temporal representations. To our knowledge, this is the first work extending GPs to model distributions over vector sets, enabling interpretable, structure-aware tracking and visualization of cross-temporal distributional shifts. Evaluated on crime hotspot migration and semantic evolution tasks, our approach significantly improves robustness and interpretability in detecting dynamic patterns. The framework establishes a novel paradigm for time-series analysis of distributional data.

Technology Category

• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning
• Data Mining & Knowledge Management: Mining of Spatial, Temporal or Spatio-Temporal Data
• Machine Learning: Time-Series/Data Streams

Application Category

• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution
• User Modeling, Personalization and Recommendation: Attacks and countermeasures in recommendation systems

Abstract

Understanding the temporal evolution of sets of vectors is a fundamental challenge across various domains, including ecology, crime analysis, and linguistics. For instance, ecosystem structures evolve due to interactions among plants, herbivores, and carnivores; the spatial distribution of crimes shifts in response to societal changes; and word embedding vectors reflect cultural and semantic trends over time. However, analyzing such time-varying sets of vectors is challenging due to their complicated structures, which also evolve over time. In this work, we propose a novel method for modeling the distribution underlying each set of vectors using infinite-dimensional Gaussian processes. By approximating the latent function in the Gaussian process with Random Fourier Features, we obtain compact and comparable vector representations over time. This enables us to track and visualize temporal transitions of vector sets in a low-dimensional space. We apply our method to both sociological data (crime distributions) and linguistic data (word embeddings), demonstrating its effectiveness in capturing temporal dynamics. Our results show that the proposed approach provides interpretable and robust representations, offering a powerful framework for analyzing structural changes in temporally indexed vector sets across diverse domains.

Problem

• Model temporal evolution of vector sets across domains
• Track structural changes in time-varying vector distributions
• Visualize transitions of vector sets in low-dimensional space

Innovation

• Gaussian processes model vector set distributions
• Random Fourier Features approximate latent functions
• Low-dimensional space tracks temporal transitions