Efficient Convex Completion of Coupled Tensors using Coupled Nuclear Norms
Coupled norms have emerged as a convex method to solve coupled tensor completion. A limitation with coupled norms is that they only induce low-rankness using the multilinear rank of coupled tensors. In this paper, we introduce a new set of coupled norms known as coupled nuclear norms by constraining the CP rank of coupled tensors. We propose new coupled completion models using the coupled nuclear norms as regularizers, which can be optimized using computationally efficient optimization methods. We derive excess risk bounds for proposed coupled completion models and show that proposed norms lead to better performance. Through simulation and real-data experiments, we demonstrate that proposed norms achieve better performance for coupled completion compared to existing coupled norms.
The paper studies the problem of coupled tensor completion, which is originally motivated from multi-task settings. The proposed method, coupled nuclear norm, extends the previously proposed tensor nuclear norm to the setting where two tensors coupled in some mode-k. Such an extension is easy to think of and does not need a lot of effort. The authors show that, using the proposed coupled nuclear norm, the risk bound is reduced, saying from O(n^0.5(K-1)) to O(n^0.5 log(n)^K); this is a distinct improvement (but the criterion used to measure the risk is questionable).  Comments:  1. The proposed method as well as analysis techniques are some kind of mixture of existing papers. Thus, the innovation level of this paper is not high. But this is okay to me, as long as the authors can make a solid contribution to the community.   2. The major concern is about the clarify, in particular the confusing notations used throughout the paper. a) As the tensors are coupled in only one mode, it is necessary to use a superscript a to specify the coupled mode. Also, it is not a good idea to begin the method with some complicated notation with subscripts (b,c,d). I think it is beneficial to keep the notations as simple as possible.  b) The proposed coupled nuclear norm defined in Equation (1) seems a set, not a norm. In particular, what are \mathcal{X} and \mathcal{Y}? The are similar problems with Equation (2).  c) On the theoretical analysis, it is really wired to split the observed entries into a training set and a test set. Usually, the test data should be the missing entries.   d) The Rademacher complexity measures only the richness of the proposed norm. However, in general, more freedoms do not necessarily lead to good recovery performance. It will be more convincing if the authors can prove that the proposed coupled nuclear norm can reduce the sampling complexity required to restore the missing entries.