High-order sum-of-squares structured tensors: theory and applications

Tensor decomposition is an important research area with numerous applications in data mining and computational neuroscience. An important class of tensor decomposition is sum-of-squares (SOS) tensor decomposition. SOS tensor decomposition has a close connection with SOS polynomials, and SOS polynomials are very important in polynomial theory and polynomial optimization. In this paper, we give a detailed survey on recent advances of high-order SOS tensors and their applications. It first shows that several classes of symmetric structured tensors available in the literature have SOS decomposition in the even order symmetric case. Then, the SOS-rank for tensors with SOS decomposition and the SOS-width for SOS tensor cones are established. Further, a sharper explicit upper bound of the SOS-rank for tensors with bounded exponent is provided, and the exact SOS-width for the cone consists of all such tensors with SOS decomposition is identified. Some potential research directions in the future are also listed in this paper.