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 semi-definite 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.

中文翻译：

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通过最大核范数复合优化完成 1 位张量补全


随着各种张量数据的出现，一位测量的张量补全作为一种基本的逆问题受到了广泛的关注。由于张量秩是许多张量数据内在结构的关键度量，并且其定义尚不唯一，因此人们提出了许多张量秩的凸代理来解决该问题，其具有计算易处理性和可靠的理论保证的优点。本文通过近似变换后的三阶张量中每个额叶的低秩性，引入了一种新的张量最大范数，并讨论了其高阶扩展。然后，对于一位张量完成，提出了与所提出的张量最大范数相关的估计器和涉及张量最大范数和张量核范数之间的混合的估计器，其中第一个估计器可以被认为是第二个估计器。还为两个估计器的恢复误差建立了上限的统计分析。理论结果表明，第二个估计量的上界优于第一个估计量，但存在阶数差距$\mathcal{O}\big{(}\sqrt{\log((n_{1}+n_{2})n_{3})}\big{)}$ 。此外，还提供了最坏情况估计器的恢复误差下界，以表明这两个估计器几乎是阶次最优的。此外，还开发了一种基于交替方向乘子法（ADMM）和半定规划（SDP）的算法来求解估计模型。通过模拟实验和推荐系统的实际应用验证了该方法的有效性。