1-Bit Tensor Completion via Max-and-Nuclear-Norm Composite Optimization

With the emergence of various tensor data, tensor completion from one-bit measurements has received widespread attention as a fundamental inverse problem. Since tensor rank is a crucial measure of the intrinsic structure in many tensor data and its definition is not yet unique, many convex surrogates of tensor rank have been proposed to solve this problem, which owns the merits of computational tractability and reliable theoretical guarantees. In this paper, a novel tensor max-norm is introduced by approximating low-rankness of each frontal slice in a transformed 3-order tensor, and its high-order extension is also discussed. Then, for one-bit tensor completion, an estimator related to the proposed tensor max-norm and another estimator involving the hybrid between tensor max-norm and tensor nuclear-norm are presented, where the first estimator can be considered as a special case of the second estimator. The statistical analysis of upper bounds is also established for recovery error of the two estimators. The theoretical results indicate that the upper bound of the second estimator is superior to the first one with the gap of order $\mathcal{O}\big(\sqrt{\log((n_1+n_2)n_3)}\big)$. In addition, a lower bound of recovery error of the worst-case estimator is provided to show that the two estimators are nearly order-optimal. Furthermore, an algorithm based on the alternating direction multipliers method (ADMM) and semidefinite programming (SDP) is developed to solve the estimation models. The effectiveness of the proposed approach is verified through the simulated experiments and a practical application in recommender-system.