Scaled Nuclear Norm Minimization for Low-Rank Tensor Completion

Minimizing the nuclear norm of a matrix has been shown to be very efficient in reconstructing a low-rank sampled matrix. Furthermore, minimizing the sum of nuclear norms of matricizations of a tensor has been shown to be very efficient in recovering a low-Tucker-rank sampled tensor. In this paper, we propose to recover a low-TT -rank sampled tensor by minimizing a weighted sum of nuclear norms of unfoldings of the tensor. We provide numerical results to show that our proposed method requires significantly less number of samples to recover to the original tensor in comparison with simply minimizing the sum of nuclear norms since the structure of the unfoldings in the TT tensor model is fundamentally different from that of matricizations in the Tucker tensor model. Tensors are generalizations of vectors and matrices to higher dimensions. Due to the recent advancement in machine learning, multidimensional analysis of data has become indispensable to fully exploit the high-dimensional representation of data as the conventional matrix analysis has only limited capability in exploiting correlations across different attributes in a multi-way representation.