PUlasso: High-Dimensional Variable Selection With Presence-Only Data

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 article 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 article, we develop the PUlasso algorithm for variable selection and classification with positive and unlabeled responses. Our algorithm involves using the majorization-minimization 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 outperforms 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. Supplementary materials for this article are available online.

Keywords:

• Majorization-minimization
• Nonconvexity
• PU-learning
• Regularization

Notes

1 We note that the group ${\ell }_1$ constraint is active only if $\sqrt{\frac{n}{\hspace{0.17em}\text{log}\hspace{0.17em}J+m}}=\mathcal{O}\left(({\max }_j{w}_j)r_0\sqrt{J}\right)$. If $R_n\ge ({\max }_j{w}_j)r_0\sqrt{J},{\Theta }_0=\{\theta ;||\theta |{|}_2\le r_0,\left|\right|\theta |{|}_{\mathit{G},2,1}\le R_n\}\supseteq \{\theta ;||\theta |{|}_2\le r_0,\left|\right|\theta |{|}_{\mathit{G},2,1}\le ({\max }_j{w}_j)r_0\sqrt{J}\}\supseteq \{\theta ;||\theta |{|}_2\le r_0\}$ by the ${\ell }_1$-${\ell }_2$ inequality, that is, if $\left|\right|\theta |{|}_2\le r_0$, $\left|\right|\theta |{|}_{\mathit{G},2,1}\le ({\max }_j{w}_j)r_0\sqrt{J}$. The other direction is trivial, and thus ${\Theta }_0$ is reduced to ${\Theta }_0=\{\theta ;||\theta |{|}_2\le r_0\}$.

2 Available at http://www.ms.k.u-tokyo.ac.jp/software.html

3 The raw data is available in https://github.com/RomeroLab/seq-fcn-data.git

Additional information

Funding

Both HS and GR were partially supported by NSF-DMS 1407028. GR was also partially supported by ARO W911NF-17-1-0357.