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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.

Long-term electroencephalogram (EEG) monitoring is effective for epilepsy diagnosis. However, it also takes a lot of time for clinicians to correctly interpret the long-time recordings. Real-time computer-based EEG monitoring and classification systems have attracted recently the attention of researchers to help clinicians locate online possible epileptic-form EEG signals. In this paper, we first present an accurate and fast EEG classification algorithm that can recognize three types of EEG signals: normal, spike, and seizure. 16-channel bipolar EEG recordings of epilepsy patients are preprocessed, segmented, and ensemble empirical mode decomposed (EEMD) into intrinsic mode functions (IMFs). Features are extracted and linear discriminant analysis (LDA) is applied to train two classifiers: one is for seizure and non-seizure discrimination, and the other is for normal and spike discrimination. In order to furthermore help the clinicians, the results of LDA are visualized and sonified. The changes of the discriminant in the LDA on continuous EEG segments are backtracked to each feature, and thus to each EEG channel. Accordingly, contours of the changes in EEG channels are depicted. At the same time, sinusoidal waves in 440 or 880 Hz are played when EEG segments are classified into spike or seizure respectively. In the experiment, EEG recordings of six subjects (two normal and four seizure patients) are examined. The experiment result shows that the accuracy of the proposed epileptic EEG classification algorithm is relatively high. In addition, the visualization and sonification algorithms of epileptic-form EEG may greatly help clinicians localize the focus of seizure and nurses take care of seizure patients, immediately.