This paper introduces the functional tensor singular value decomposition (FTSVD), a novel dimension reduction framework for tensors with one functional mode and several tabular modes. The problem is motivated by high-order longitudinal data analysis. Our model assumes the observed data to be a random realization of an approximate CP low-rank functional tensor measured on a discrete time grid. Incorporating tensor algebra and the theory of Reproducing Kernel Hilbert Space (RKHS), we propose a novel RKHS-based constrained power iteration with spectral initialization. Our method can successfully estimate both singular vectors and functions of the low-rank structure in the observed data. With mild assumptions, we establish the non-asymptotic contractive error bounds for the proposed algorithm. The superiority of the proposed framework is demonstrated via extensive experiments on both simulated and real data.

翻译：本文提出了功能张量奇异值分解（FTSVD），这是一个针对含有一个功能模式和多个表格模式的张量的新型降维框架。该问题源于高阶纵向数据分析。我们的模型假设观测数据是近似CP低秩功能张量在离散时间网格上测量的随机实现。结合张量代数与再生核希尔伯特空间（RKHS）理论，我们提出了一种基于RKHS约束的谱初始化幂迭代方法。该方法能够有效估计观测数据中低秩结构的奇异向量与函数。在温和假设下，我们建立了所提算法的非渐近收缩误差界。通过模拟和真实数据的广泛实验，验证了所提框架的优越性。