Generalized high-dimensional trace regression via nuclear norm regularization

Jianqing Fan Wenyan Gong Ziwei Zhu

Optimization and Control mathscidoc:1912.43417

Journal of Econometrics, 2019.5
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 ( 1 Y + b ()), where = tr ( T X). 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 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
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@inproceedings{jianqing2019generalized,
  title={Generalized high-dimensional trace regression via nuclear norm regularization},
  author={Jianqing Fan, Wenyan Gong, and Ziwei Zhu},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20191221114312938142977},
  booktitle={Journal of Econometrics},
  year={2019},
}
Jianqing Fan, Wenyan Gong, and Ziwei Zhu. Generalized high-dimensional trace regression via nuclear norm regularization. 2019. In Journal of Econometrics. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20191221114312938142977.
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