稳健矩阵回归模型和方法研究

随着大数据时代的到来,我们面临的数据越来越复杂,其中待估系数为矩阵的模型亟待构造和求解.无论在统计还是优化领域,许多专家学者都致力于矩阵模型的统计性质分析及寻找其最优解的算法设计.当随机误差期望为0且同方差时,采用基于最小二乘的模型可以很好地解决问题.当随机误差异方差,分布为重尾分布(如双指数分布,t-分布等)或数据含有异常值时,需要考虑稳健的方法来求解问题.常用的稳健方法有最小一乘,分位数,Huber等.目前稳健方法的研究大多集中于线性回归问题,对于矩阵回归问题的研究比较缺乏.本文从最小二乘模型讲起,对矩阵回归问题进行了总结和评述,同时列出了一些文献和简要介绍了我们的近期的部分工作.最后对于稳健矩阵回归,我们提出了一些展望和设想.

THE STUDY OF ROBUST MATRIX REGRESSION MODELS AND ALGORITHMS

With the breakout of the big data era, the data we are interested in is more and more complex. Especially, the model with matrix coefficient is in urgent to be constructed and solved. Many scholars are devoted to studying the statistical property analysis and designing algorithms for solving the matrix model. When the random errors have expectation 0 and the same variance, the method based on the least square loss function may perform well. However, when the random errors are heteroscedastic or the distribution of the errors are heavy-tailed (such as bi-exponential distribution, t-distribution, etc.) or the data contain outliers, the robust methods should be considered. The common robust methods are LAD, Quantile, Huber, etc. Most of the current research on the robust methods focus on the linear regression problem. There is few research on matrix regression problem. In this paper, we start with the least squares models, then summarize and comment on some matrix regression models. At the same time, we list some papers and briefly introduce some of our recent work. Finally, for the robust matrix regression problem, we propose some ideas and prospects.