  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 $f(Y\vert X)=c(Y)\exp{(\phi^{-1}\left[-Y\eta^* + b(\eta^*)\right])}$,
where $\eta^*=tr({\Theta^*}^TX)$. 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 $\Theta^*$ 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.
