# Logistic principal component analysis via non-convex singular value thresholding

## TL;DR

This paper introduces a non-convex singular value thresholding approach for logistic PCA to better recover low rank structures in binary data, avoiding overfitting and improving model selection and performance.

## Contribution

It proposes a novel non-convex thresholding method for logistic PCA, along with an efficient algorithm and cross-validation procedure for model selection, outperforming existing methods.

## Key findings

- CV-based model selection effectively identifies the proposed model.
- The method outperforms convex nuclear norm and exact low rank models.
- Successful application to biological binary data demonstrates practical utility.

## Abstract

Multivariate binary data is becoming abundant in current biological research. Logistic principal component analysis (PCA) is one of the commonly used tools to explore the relationships inside a multivariate binary data set by exploiting the underlying low rank structure. We re-expressed the logistic PCA model based on the latent variable interpretation of the generalized linear model on binary data. The multivariate binary data set is assumed to be the sign observation of an unobserved quantitative data set, on which a low rank structure is assumed to exist. However, the standard logistic PCA model (using exact low rank constraint) is prone to overfitting, which could lead to divergence of some estimated parameters towards infinity. We propose to fit a logistic PCA model through non-convex singular value thresholding to alleviate the overfitting issue. An efficient Majorization-Minimization algorithm is implemented to fit the model and a missing value based cross validation (CV) procedure is introduced for the model selection. Our experiments on realistic simulations of imbalanced binary data and low signal to noise ratio show that the CV error based model selection procedure is successful in selecting the proposed model. Furthermore, the selected model demonstrates superior performance in recovering the underlying low rank structure compared to models with convex nuclear norm penalty and exact low rank constraint. A binary copy number aberration data set is used to illustrate the proposed methodology in practice.