  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.
