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Motivated by the Tucker decomposition, in this paper we introduce a new tensor decomposition for third order tensors, which decomposes a third order tensor to three third order factor tensors. Each factor tensor has two low dimensions. We call such a decomposition the triple decomposition, and the corresponding rank the triple rank. The triple rank of a third order tensor is not greater than the middle value of the Tucker rank. The number of parameters in the bilevel form of standard triple decomposition is less than the number of parameters of Tucker decomposition in substantial cases. The theoretical discovery is confirmed numerically. Numerical tests show that third order tensor data from practical applications such as internet traffic and image are of low triple ranks. A tensor recovery method based on low rank triple decomposition is proposed. Its convergence and convergence rate are established. Numerical experiments confirm the efficiency of this method. [ABSTRACT FROM AUTHOR]

Comon's Conjecture claims that for a symmetric tensor, its rank and its symmetric rank coincide. We show that this conjecture is true under an additional assumption that the rank of that tensor is not larger than its order. Moreover, if its rank is less than its order, then all rank decompositions are necessarily symmetric rank decompositions. [ABSTRACT FROM AUTHOR]

In this paper, we show that for a symmetric tensor, its best symmetric rank-1 approximation is its best rank-1 approximation. Based on this result, a positive lower bound for the best rank-1 approximation ratio of a symmetric tensor is given. Furthermore, a higher order polynomial spherical optimization problem can be reformulated as a multilinear spherical optimization problem. Then, we present a modified power algorithm for solving the homogeneous polynomial spherical optimization problem. Numerical results are presented, illustrating the effectiveness of the proposed algorithm. [ABSTRACT FROM AUTHOR]