PULasso: High-dimensional variable selection with presence-only data

TL;DR

This paper introduces PUlasso, an algorithm for variable selection and classification in high-dimensional presence-only data, addressing statistical and computational challenges with theoretical guarantees and superior performance in simulations and real data.

Contribution

We develop PUlasso, a scalable algorithm using MM framework for variable selection in presence-only data, with convergence guarantees and minimax optimal error bounds.

Findings

PUlasso outperforms existing algorithms in moderate p settings.

Theoretical convergence to stationary points with optimal error bounds.

Effective in real biochemistry data analysis.

Abstract

In various real-world problems, we are presented with classification problems with positive and unlabeled data, referred to as presence-only responses. In this paper, we study variable selection in the context of presence only responses where the number of features or covariates p is large. The combination of presence-only responses and high dimensionality presents both statistical and computational challenges. In this paper, we develop the PUlasso algorithm for variable selection and classification with positive and unlabeled responses. Our algorithm involves using the majorization-minimization (MM) framework which is a generalization of the well-known expectation-maximization (EM) algorithm. In particular to make our algorithm scalable, we provide two computational speed-ups to the standard EM algorithm. We provide a theoretical guarantee where we first show that our algorithm converges to a stationary point, and then prove that any stationary point within a local neighborhood of the true parameter achieves the minimax optimal mean-squared error under both strict sparsity and group sparsity assumptions. We also demonstrate through simulations that our algorithm out-performs state-of-the-art algorithms in the moderate p settings in terms of classification performance. Finally, we demonstrate that our PUlasso algorithm performs well on a biochemistry example.