Generalized high-dimensional trace regression via nuclear norm regularization
Generalized high-dimensional trace regression via nuclear norm regularization
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DOI：
10.1016/j.jeconom.2019.04.026
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发表时间：
2019-09-01
影响因子：
6.3
通讯作者：
Zhu, Ziwei
中科院分区：
文献类型：
--
作者：
Fan, Jianqing;Gong, Wenyan;Zhu, Ziwei
本文研究了回归系数矩阵近似为低秩的广义迹回归，推广了回归系数向量稀疏性的概念。具体而言，给定矩阵协变量X，响应Y的概率密度函数为f（Y| X）=c（Y）exp（X-1-Y*+B（？*）），其中phi=tr（T*TX）。该模型可容纳各种类型的响应，并包含许多重要的问题设置，例如降秩回归，矩阵回归，可容纳一组回归量，矩阵完成等。我们通过最小化经验负对数似然加上核范数惩罚来估计T*。我们首先建立了一个一般的理论，然后对于每个特定的问题，我们推导出明确的统计率的估计。它们都匹配线性迹回归中的极小极大率，直到对数因子。数值研究证实了我们建立的速率，并证明了广义迹回归的优势，线性迹回归时的响应是二分的。我们还展示了将核范数正则化用于动态股票收益预测和图像分类的好处。(C)2019爱思唯尔B. V.保留所有权利。
We study the generalized trace regression with a near low-rank regression coefficient matrix, which extends notion of sparsity for regression coefficient vectors. Specifically, given a matrix covariate X, the probability density function of the response Y is f(Y|X)=c(Y)exp(X-1-Y*+b(?*)), where phi=tr(T*TX). This model accommodates various types of responses and embraces many important problem setups such as reduced-rank regression, matrix regression that accommodates a panel of regressors, matrix completion, among others. We estimate T* through minimizing empirical negative log-likelihood plus nuclear norm penalty. We first establish a general theory and then for each specific problem, we derive explicitly the statistical rate of the proposed estimator. They all match the minimax rates in the linear trace regression up to logarithmic factors. Numerical studies confirm the rates we established and demonstrate the advantage of generalized trace regression over linear trace regression when the response is dichotomous. We also show the benefit of incorporating nuclear norm regularization in dynamic stock return prediction and in image classification. (C) 2019 Elsevier B.V. All rights reserved.