Fast multilinear singular value decomposition for structured tensors

The higher-order singular value decomposition (HOSVD) is a generalization of the singular value decomposition (SVD) to higher-order tensors (i.e., arrays with more than two indices) and plays an important role in various domains. Unfortunately, this decomposition is computationally demanding. Indeed, the HOSVD of a third-order tensor involves the computation of the SVD of three matrices, which are referred to as "modes" or "matrix unfoldings." In this paper, we present fast algorithms for computing the full and the rank-truncated HOSVD of third-order structured (symmetric, Toeplitz, and Hankel) tensors. These algorithms are derived by considering two specific ways to unfold a structured tensor, leading to structured matrix unfoldings whose SVD can be efficiently computed.