Multiway datasets are commonly analyzed using unsupervised matrix and tensor factorization methods to reveal underlying patterns. Frequently, such datasets include timestamps and could correspond to, for example, health-related measurements of subjects collected over time. The temporal dimension is inherently different from the other dimensions, requiring methods that account for its intrinsic properties. Linear Dynamical Systems (LDS) are specifically designed to capture sequential dependencies in the observed data. In this work, we bridge the gap between tensor factorizations and dynamical modeling by exploring the relationship between LDS, Coupled Matrix Factorizations (CMF) and the PARAFAC2 model. We propose a time-aware coupled factorization model called d(ynamical)CMF that constrains the temporal evolution of the latent factors to adhere to a specific LDS structure. Using synthetic datasets, we compare the performance of dCMF with PARAFAC2 and t(emporal)PARAFAC2 which incorporates temporal smoothness. Our results show that dCMF and PARAFAC2-based approaches perform similarly when capturing smoothly evolving patterns that adhere to the PARAFAC2 structure. However, dCMF outperforms alternatives when the patterns evolve smoothly but deviate from the PARAFAC2 structure. Furthermore, we demonstrate that the proposed dCMF method enables to capture more complex dynamics when additional prior information about the temporal evolution is incorporated.

翻译：多路数据集通常通过无监督的矩阵和张量分解方法进行分析，以揭示其潜在模式。这类数据集常包含时间戳，例如可能对应随时间收集的受试者健康相关测量值。时间维度本质上不同于其他维度，需要能够处理其内在特性的方法。线性动态系统（LDS）专为捕捉观测数据中的序列依赖性而设计。在本工作中，我们通过探索LDS、耦合矩阵分解（CMF）与PARAFAC2模型之间的关系，弥合了张量分解与动态建模之间的鸿沟。我们提出了一种称为d（动态）CMF的时间感知耦合分解模型，该模型约束潜在因子的时序演化，使其遵循特定的LDS结构。使用合成数据集，我们将dCMF与PARAFAC2以及引入时间平滑性的t（时序）PARAFAC2的性能进行了比较。结果表明，在捕捉符合PARAFAC2结构的平滑演化模式时，dCMF与基于PARAFAC2的方法表现相似。然而，当模式平滑演化但偏离PARAFAC2结构时，dCMF优于其他方法。此外，我们证明当纳入更多关于时序演化的先验信息时，所提出的dCMF方法能够捕捉更复杂的动态特性。