一类基于L0/1软间隔损失函数的低秩支持张量机

支持向量机作为基于向量空间的一种传统的机器学习方法，不能直接处理张量类型的数据，否则不仅破坏数据的空间结构，还会造成维度灾难及小样本问题。作为支持向量机的一种高阶推广，用于处理张量数据分类的支持张量机已经引起众多学者的关注，并应用于遥感成像、视频分析、金融、故障诊断等多个领域。与支持向量机类似，已有的支持张量机模型中采用的损失函数多为L_0/1函数的代理函数。将直接使用L_0/1这一本原函数作为损失函数，并利用张量数据的低秩性，建立针对二分类问题的低秩支持张量机模型。针对这一非凸非连续的张量优化问题，设计交替方向乘子法进行求解，并通过对模拟数据和真实数据进行数值实验，验证模型与算法的有效性。

Low rank support tensor machine based on L0/1 soft-margin loss function

As a traditional classification method, support vector machine (SVM) has limitations for high order tensorial data, since direct vectorization will lead to the loss of intrinsic spatial structures in tensors, and the small sample size problem as well. As a higher-order extension of SVM, support tensor machine (STM), which targets at tensorial data classification, has attracted more and more attention of many scholars, with wide applications in remote sensing imaging, video processing, finance, fault diagnosis, etc. Analogous to SVM, the involved loss functions in most of the existing STM models are surrogates of the L_0/1 function. In this paper, the original L_0/1 loss is employed, based on which, a low rank STM model is proposed for the binary classification problem, with consideration of the intrinsic low-rankness of tensorial data. The resulting nonconvex discontinuous tensor optimization problem is solved by an alternating direction method of multipliers. Numerical experiments are conducted on synthetic data and real data sets to demonstrate the effectiveness of the proposed approach.