Learning Efficient Tensor Representations with Ring Structure Networks
For machine learning practitioners using tensor decompositions, this work provides a more robust and expressive representation that handles dimension permutation, though it is an incremental improvement over existing TT methods.
The paper proposes a tensor ring (TR) representation that is permutation symmetric, addressing the inconsistency of tensor train (TT) decomposition under dimension permutation. Experiments show TR networks are more expressive and informative than TT networks on synthetic and real-world data.
Tensor train (TT) decomposition is a powerful representation for high-order tensors, which has been successfully applied to various machine learning tasks in recent years. However, since the tensor product is not commutative, permutation of data dimensions makes solutions and TT-ranks of TT decomposition inconsistent. To alleviate this problem, we propose a permutation symmetric network structure by employing circular multilinear products over a sequence of low-order core tensors. This network structure can be graphically interpreted as a cyclic interconnection of tensors, and thus we call it tensor ring (TR) representation. We develop several efficient algorithms to learn TR representation with adaptive TR-ranks by employing low-rank approximations. Furthermore, mathematical properties are investigated, which enables us to perform basic operations in a computationally efficiently way by using TR representations. Experimental results on synthetic signals and real-world datasets demonstrate that the proposed TR network is more expressive and consistently informative than existing TT networks.