TY - JOUR

T1 - Generalized high-dimensional trace regression via nuclear norm regularization

AU - Fan, Jianqing

AU - Gong, Wenyan

AU - Zhu, Z.

N1 - Funding Information:
This paper is supported by NSF grants DMS-1406266, DMS-1662139, and DMS-1712591. ? This paper is supported by NSF grants DMS-1406266, DMS-1662139, and DMS-1712591.

PY - 2019/9

Y1 - 2019/9

N2 - 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(ϕ−1−Yη∗+b(η∗)), where η∗=tr(Θ∗ 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 Θ∗ 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.

AB - 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(ϕ−1−Yη∗+b(η∗)), where η∗=tr(Θ∗ 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 Θ∗ 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.

KW - High dimensional statistics

KW - Logistic regression

KW - Matrix completion

KW - Nuclear norm regularization

KW - Restricted strong convexity

KW - Trace regression

UR - http://www.scopus.com/inward/record.url?scp=85065133913&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85065133913&partnerID=8YFLogxK

U2 - 10.1016/j.jeconom.2019.04.026

DO - 10.1016/j.jeconom.2019.04.026

M3 - Article

AN - SCOPUS:85065133913

VL - 212

SP - 177

EP - 202

JO - Journal of Econometrics

JF - Journal of Econometrics

SN - 0304-4076

IS - 1

ER -