Simultaneous Feature Selection and Outlier Detection with Optimality Guarantees

Abstract

Sparse estimation methods capable of tolerating outliers have been broadly investigated in the last decade. We contribute to this research considering high-dimensional regression problems contaminated by multiple mean-shift outliers which affect both the response and the design matrix. We develop a general framework for this class of problems and propose the use of mixed-integer programming to simultaneously perform feature selection and outlier detection with provably optimal guarantees. We characterize the theoretical properties of our approach, i.e. a necessary and sufficient condition for the robustly strong oracle property, which allows the number of features to exponentially increase with the sample size; the optimal estimation of the parameters; and the breakdown point of the resulting estimates. Moreover, we provide computationally efficient procedures to tune integer constraints and to warm-start the algorithm. We show the superior performance of our proposal compared to existing heuristic methods through numerical simulations and an application investigating the relationships between the human microbiome and childhood obesity.

References in corpus (9)

• Nearly unbiased variable selection under minimax concave penalty
• Strong oracle optimality of folded concave penalized estimation
• Sparse least trimmed squares regression for analyzing high-dimensional large data sets
• Propagation of outliers in multivariate data
• Penalized Composite Quasi-Likelihood for Ultrahigh-Dimensional Variable Selection
• Extended Comparisons of Best Subset Selection, Forward Stepwise Selection, and the Lasso
• Least quantile regression via modern optimization
• Regularization of Case-Specific Parameters for Robustness and Efficiency
• MIP-BOOST: Efficient and Effective $L_0$ Feature Selection for Linear Regression

Cited by in corpus (2)

• Robust Variable Selection and Estimation Via Adaptive Elastic Net S-Estimators for Linear Regression
• Doubly Robust Feature Selection with Mean and Variance Outlier Detection and Oracle Properties