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.

翻译：理解向量集的时序演化是生态学、犯罪分析及语言学等多个领域的基础性挑战。例如，生态系统结构因植物、食草动物与食肉动物间的相互作用而演变；犯罪空间分布随社会变迁发生转移；词嵌入向量则随时间反映文化与语义趋势。然而，由于这些向量集结构复杂且随时间动态变化，分析其时变特性具有挑战性。本研究提出一种新方法，利用无限维高斯过程对每个向量集背后的分布进行建模。通过随机傅里叶特征逼近高斯过程中的隐函数，我们获得了随时间变化的紧凑且可比较的向量表示。这使得我们能够在低维空间中追踪并可视化向量集的时序变迁。我们将该方法应用于社会学数据（犯罪分布）和语言学数据（词嵌入），验证了其在捕捉时序动态方面的有效性。结果表明，所提方法提供了可解释且稳健的表示，为跨领域时序索引向量集的结构变化分析提供了强大框架。