# Generalized High-Dimensional Trace Regression via Nuclear Norm Regularization

arXiv: https://arxiv.org/abs/1710.08083 · HTML: https://mlsanity.com/paper/1710.08083/

## Key facts

- Cited by 12 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 23 Oct 2017.

## Built on by (most cited first, 12 of 12)

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- [Nonparametric Trace Regression in High Dimensions via Sign Series Representation](https://mlsanity.com/paper/2105.01783/) (2021)
- [A Unified Framework for Estimation of High-dimensional Conditional Factor Models](https://mlsanity.com/paper/2209.00391/) (2022)
- [Computational Efficient and Minimax Optimal Nonignorable Matrix Completion](https://mlsanity.com/paper/2504.04016/) (2025)
- [Robust Tensor Regression with Nonconvexity: Algorithmic and Statistical Theory](https://mlsanity.com/paper/2605.07448/) (2026)

## Abstract (arXiv)

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(Yvert X)=c(Y)exp(φ-1left[-Yη^* + b(η^*)right]), where η^*=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.