Projective Inference in High-dimensional Problems: Prediction and Feature Selection
This work addresses the problem of feature selection and prediction in high-dimensional, scarce data settings for researchers and practitioners in statistics and machine learning, offering incremental improvements by unifying existing techniques and providing new evaluation methods.
The paper tackles predictive inference and feature selection in high-dimensional generalized linear models by proposing a two-stage approach that first builds a predictive reference model and then projects it to a minimal feature subset, achieving a tradeoff between sparsity and accuracy. It introduces a new projection technique, a fast cross-validation method for model selection, and a theoretical analysis, with benefits demonstrated through simulated and real-world examples.
This paper discusses predictive inference and feature selection for generalized linear models with scarce but high-dimensional data. We argue that in many cases one can benefit from a decision theoretically justified two-stage approach: first, construct a possibly non-sparse model that predicts well, and then find a minimal subset of features that characterize the predictions. The model built in the first step is referred to as the \emph{reference model} and the operation during the latter step as predictive \emph{projection}. The key characteristic of this approach is that it finds an excellent tradeoff between sparsity and predictive accuracy, and the gain comes from utilizing all available information including prior and that coming from the left out features. We review several methods that follow this principle and provide novel methodological contributions. We present a new projection technique that unifies two existing techniques and is both accurate and fast to compute. We also propose a way of evaluating the feature selection process using fast leave-one-out cross-validation that allows for easy and intuitive model size selection. Furthermore, we prove a theorem that helps to understand the conditions under which the projective approach could be beneficial. The benefits are illustrated via several simulated and real world examples.