Generalized high-dimensional trace regression via nuclear norm regularization

Jianqing Fan, Wenyan Gong, Z. Zhu

Research output: Contribution to journalArticle

3 Scopus citations

Abstract

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.

Original languageEnglish (US)
Pages (from-to)177-202
Number of pages26
JournalJournal of Econometrics
Volume212
Issue number1
DOIs
StatePublished - Sep 2019

All Science Journal Classification (ASJC) codes

  • Economics and Econometrics

Keywords

  • High dimensional statistics
  • Logistic regression
  • Matrix completion
  • Nuclear norm regularization
  • Restricted strong convexity
  • Trace regression

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