# Robust Sparse Reduced Rank Regression in High Dimensions

arXiv: https://arxiv.org/abs/1810.07913 · HTML: https://mlsanity.com/paper/1810.07913/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 18 Oct 2018.

## Built on by (most cited first, 1 of 1)

- [Mining the Factor Zoo: Estimation of Latent Factor Models with Sufficient Proxies](https://mlsanity.com/paper/2212.12845/) (2022)

## Abstract (arXiv)

We propose robust sparse reduced rank regression for analyzing large and complex high-dimensional data with heavy-tailed random noise. The proposed method is based on a convex relaxation of a rank- and sparsity-constrained non-convex optimization problem, which is then solved using the alternating direction method of multipliers algorithm. We establish non-asymptotic estimation error bounds under both Frobenius and nuclear norms in the high-dimensional setting. This is a major contribution over existing results in reduced rank regression, which mainly focus on rank selection and prediction consistency. Our theoretical results quantify the tradeoff between heavy-tailedness of the random noise and statistical bias. For random noise with bounded (1+δ)th moment with δ ∈ (0,1), the rate of convergence is a function of δ, and is slower than the sub-Gaussian-type deviation bounds; for random noise with bounded second moment, we obtain a rate of convergence as if sub-Gaussian noise were assumed. Furthermore, the transition between the two regimes is smooth. We illustrate the performance of the proposed method via extensive numerical studies and a data application.