Functional Tensor Singular Value Decomposition

Multidimensional data can naturally be represented by tensors. However, due to the inherent discrete nature, the classic tensor representation is unsuitable for challenging missing slice recovery tasks along the third mode in real-world applications, e.g., traffic data imputation, hyperspectral band recovery, and video frame interpolation. As an alternative to the classic discrete tensor representation, we suggest a matrix-valued function representation, which can continuously represent data along the third mode. For the suggested function representation, we develop fundamental functional tensor singular value decomposition (FT-SVD) and establish the best-rank-k approximation theorem for the induced functional rank. FT-SVD, which can ingeniously capture the low-rankness and smoothness underlying the data, has the ability to address challenging missing slice recovery tasks. Based on FT-SVD, we propose a multidimensional data completion model and develop the optimization algorithm with a convergence guarantee. Numerical results on synthetic and real data demonstrate the superiority of FT-SVD over discrete tensor SVD, especially for the missing slice recovery tasks in real-world applications.