Tensor decomposition is an important tool for multiway data analysis. In practice, the data is often sparse yet associated with rich temporal information. Existing methods, however, often under-use the time information and ignore the structural knowledge within the sparsely observed tensor entries. To overcome these limitations and to better capture the underlying temporal structure, we propose Dynamic EMbedIngs fOr dynamic Tensor dEcomposition (DEMOTE). We develop a neural diffusion-reaction process to estimate dynamic embeddings for the entities in each tensor mode. Specifically, based on the observed tensor entries, we build a multi-partite graph to encode the correlation between the entities. We construct a graph diffusion process to co-evolve the embedding trajectories of the correlated entities and use a neural network to construct a reaction process for each individual entity. In this way, our model can capture both the commonalities and personalities during the evolution of the embeddings for different entities. We then use a neural network to model the entry value as a nonlinear function of the embedding trajectories. For model estimation, we combine ODE solvers to develop a stochastic mini-batch learning algorithm. We propose a stratified sampling method to balance the cost of processing each mini-batch so as to improve the overall efficiency. We show the advantage of our approach in both simulation study and real-world applications. The code is available at https://github.com/wzhut/Dynamic-Tensor-Decomposition-via-Neural-Diffusion-Reaction-Processes.

翻译：张量分解是多维数据分析的重要工具。实际数据通常稀疏且伴随丰富的时间信息，但现有方法往往未能充分利用时间信息并忽略稀疏观测张量条目中的结构知识。为克服这些限制并更好地捕捉潜在的时间结构，我们提出面向动态张量分解的动态嵌入方法（DEMOTE）。我们开发了一种神经扩散-反应过程来估计每个张量模式中实体的动态嵌入。具体而言，基于观测到的张量条目，我们构建了一个多部图来编码实体间的相关性。我们设计了一个图扩散过程，使相关实体的嵌入轨迹共同演化，并利用神经网络为每个实体构建独立的反应过程。通过这种方式，模型能够同时捕捉不同实体嵌入演化过程中的共性与个性。随后，我们使用神经网络将条目值建模为嵌入轨迹的非线性函数。在模型估计方面，我们结合常微分方程求解器开发了随机小批量学习算法，并提出了分层采样方法平衡每个小批量的处理成本，从而提升整体效率。模拟实验和实际应用均验证了我们方法的优势。代码已开源：https://github.com/wzhut/Dynamic-Tensor-Decomposition-via-Neural-Diffusion-Reaction-Processes。