2018/10/18 by Kean Ming Tan, Qiang Sun, Tan, Kean Ming +3
Engineering · Mathematics · #Advanced Statistical Methods and Models #FOS: Computer and information sciences #Machine Learning (cs.LG) #Machine Learning (stat.ML) #Sparse and Compressive Sensing Techniques #Statistical Methods and Inference
paper · pdf · doi:10.48550/arxiv.1810.07913
openalex publication_date 2018/10/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.