Simultaneous Feature Selection and Outlier Detection with Optimality Guarantees
Abstract
Biomedical research is increasingly data rich, with studies comprising ever growing numbers of features. The larger a study, the higher the likelihood that a substantial portion of the features may be redundant and/or contain contamination (outlying values). This poses serious challenges, which are exacerbated in cases where the sample sizes are relatively small. Effective and effcient approaches to perform sparse estimation in the presence of outliers are critical for these studies, and have received considerable attention in the last decade. We contribute to this area considering high-dimensional regressions contaminated by multiple mean-shift outliers affecting both the response and the design matrix. We develop a general framework and use mixed-integer programming to simultaneously perform feature selection and outlier detection with provably optimal guarantees. We prove theoretical properties for our approach, i.e., a necessary and suffcient condition for the robustly strong oracle property, where the number of features can increase exponentially with the sample size; the optimal estimation of parameters; and the breakdown point of the resulting estimates. Moreover, we provide computationally effcient procedures to tune integer constraints and warm-start the algorithm. We show the superior performance of our proposal compared to existing heuristic methods through simulations and use it to study the relationships between childhood obesity and the human microbiome.
Under review.
Luca Insolia
Postdoctoral Researcher
My primary research interests concern robust statistics and high-dimensional modeling. During my PhD, I developed statistical methodologies for analyzing sparse regression problems affected by different forms of adversarial data contamination. The developed methodologies encompass continuous optimization methods as well as mixed-integer programming techniques. I applied these tools to analyze biomedical data and to investigate the main possible drivers of honey bee colony loss.
Ana Kenney
Postdoc
Francesca Chiaromonte
Full Professor
