Linear regression

Quantum mechanics

Linguistics

Philosophy

Statistics

Proper linear model

Minimax

Design matrix

Applied mathematics

Regularization (linguistics)

Eigenvalues and eigenvectors

Computer science

Mathematical optimization

TRACE (psycholinguistics)

Regression analysis

Physics

Regression

Mathematics

Estimator

Econometrics

Artificial intelligence

Polynomial regression

Matrix norm

Covariate

Journal of Econometrics

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.

Generalized high-dimensional trace regression via nuclear norm regularization