基于Huber损失的稳健张量回归及其应用  

Robust Tensor Regression Based on the Huber Loss and Its Application

作　　者：李传权 马海强 刘小惠 刘育孜 LI Chuan-quan;MA Hai-qiang;LIU Xiao-hui;LIU Yu-zi(Jiangxi University of Finance and Economics,Nanchang 330013,China;Key Laboratory of Data Science in Finance and Economics,Jiangxi University of Finance and Economics,Nanchang 330013,China)

机构地区：[1]江西财经大学,江西南昌330013 [2]江西财经大学财经数据科学重点实验室,江西南昌330013

出　　处：《数理统计与管理》2024年第4期571-586,共16页Journal of Applied Statistics and Management

基　　金：国家社科重大项目基金(21&ZD152);国家自然科学基金(12161042,11971208,12301377);江西省教育厅科学技术研究项目(GJJ210535,GJJ200522,GJJ2200539);江西省科技厅基金项目(20232BAB211014,20224ACB211003);江西财经大学财经数据科学重点实验室开放课题。

摘　　要：随着科学技术的进步,张量数据及相关方法在众多领域中得到了快速的发展和广泛的运用。一系列基于CP(CANDECOMP/PARAFAC)分解的张量回归也逐渐被提出,但是在实际问题中,传统的张量回归方法易受厚尾数据、异常值等因素影响,从而造成系数估计的偏差。鉴于此,本文提出基于Huber损失的稳健张量回归以及其稀疏形式,并构造了稳健块松弛算法及其稀疏算法,对其进行优化求解。同时,本文证明了稳健张量回归中估计系数的相合性和渐近正态性,也给出了稀疏形式下回归系数的误差界。最后,模拟实验和京津冀地区PM_(2.5)数据均证实本文所提的方法比传统的张量回归具有更好的稳健性和更加精确的预测能力。With the rapid development of science and technology,tensor data and related methods have gained fast development and wide application in many fields.A series of tensor regression methods based on the CP(CANDECOMP/PARAFAC)decomposition are proposed.However,these methods are often infuenced by the heavy-tailed datasets or outliers in the real datasets,which lead to the bias of estimator coefficient.In this paper,we build the robust tensor regression and its sparse method by using the Huber loss.The robust relaxed block algorithm and its sparse type are built to optimize these two methods.Meanwhile,we prove the coefficient's consistency and asymptotically normality of robust tensor regression,and also obtain the coefficient's error bound of sparse method.Finally,the simulation datasets and the PM_(2.5) data of Beijing-Tianjin-Hebei region show that our methods are more robust and predictable than traditional tensor regression.

关 键 词：Huber损失函数 稳健性 CP分解 张量回归 PM_(2.5) 

分 类 号：O212[理学—概率论与数理统计]