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Sparse Reduced Rank Huber Regression in High Dimensions

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Version 2 2022-04-15, 14:00
Version 1 2022-03-10, 15:40
journal contribution
posted on 2022-04-15, 14:00 authored by Kean Ming Tan, Qiang Sun, Daniela Witten

We propose a sparse reduced rank Huber 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 nonconvex optimization problem, which is then solved using a block coordinate descent and an alternating direction method of multipliers algorithm. We establish nonasymptotic 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. We illustrate the performance of the proposed method via extensive numerical studies and a data application. Supplementary materials for this article are available online.


Tan is supported by NSF DMS 2113356, NSF DMS 1949730, and NIH RF1-MH122833. Sun is supported in part by NSERC grant RGPIN-2018-06484. Witten is supported by NIH R01 GM123993, Simons Investigator for Mathematical Modeling of Living Systems and NSF CAREER award 1252624.


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