Rank Properties and Computational Methods for Orthogonal Tensor Decompositions

Abstract

The orthogonal decomposition factorizes a tensor into a sum of an orthogonal list of rank-one tensors. The corresponding rank is called orthogonal rank. We present several properties of orthogonal rank, which are different from those of tensor rank in many aspects. For instance, a subtensor may have a larger orthogonal rank than the whole tensor. To fit the orthogonal decomposition, we propose an algorithm based on the augmented Lagrangian method. The gradient of the objective function has a nice structure, inspiring us to use gradient-based optimization methods to solve it. We guarantee the orthogonality by a novel orthogonalization process. Numerical experiments show that the proposed method has a great advantage over the existing methods for strongly orthogonal decompositions in terms of the approximation error.