Robust Fused Lasso Penalized Huber Regression with Nonasymptotic Property and Implementation Studies

TL;DR

This paper introduces a robust fused lasso penalized Huber regression method that effectively handles outliers and promotes local constancy in high-dimensional data, with proven nonasymptotic error bounds and demonstrated superior robustness and prediction in simulations and cancer data.

Contribution

It develops a novel adaptive Huber regression with fused lasso penalty, providing nonasymptotic error bounds and efficient optimization for robust high-dimensional analysis.

Findings

The method achieves better robustness against outliers.

It provides nonasymptotic error bounds in high-dimensional settings.

Demonstrates improved prediction accuracy on real cancer data.

Abstract

For some special data in reality, such as the genetic data, adjacent genes may have the similar function. Thus ensuring the smoothness between adjacent genes is highly necessary. But, in this case, the standard lasso penalty just doesn't seem appropriate anymore. On the other hand, in high-dimensional statistics, some datasets are easily contaminated by outliers or contain variables with heavy-tailed distributions, which makes many conventional methods inadequate. To address both issues, in this paper, we propose an adaptive Huber regression for robust estimation and inference, in which, the fused lasso penalty is used to encourage the sparsity of the coefficients as well as the sparsity of their differences, i.e., local constancy of the coefficient profile. Theoretically, we establish its nonasymptotic estimation error bounds under $\ell_2$-norm in high-dimensional setting. The proposed estimation method is formulated as a convex, nonsmooth and separable optimization problem, hence, the alternating direction method of multipliers can be employed. In the end, we perform on simulation studies and real cancer data studies, which illustrate that the proposed estimation method is more robust and predictive.