Dimensionality Reduction for Binary Data through the Projection of Natural Parameters

TL;DR

This paper introduces a new formulation of logistic PCA for binary data that avoids matrix factorization, enabling efficient computation and better scalability, demonstrated through simulations and medical data analysis.

Contribution

It proposes a novel projection-based logistic PCA method that simplifies computation and improves scalability over existing matrix factorization approaches.

Findings

The new method is computationally efficient.

It outperforms previous logistic PCA formulations.

It effectively analyzes binary medical data.

Abstract

Principal component analysis (PCA) for binary data, known as logistic PCA, has become a popular alternative to dimensionality reduction of binary data. It is motivated as an extension of ordinary PCA by means of a matrix factorization, akin to the singular value decomposition, that maximizes the Bernoulli log-likelihood. We propose a new formulation of logistic PCA which extends Pearson's formulation of a low dimensional data representation with minimum error to binary data. Our formulation does not require a matrix factorization, as previous methods do, but instead looks for projections of the natural parameters from the saturated model. Due to this difference, the number of parameters does not grow with the number of observations and the principal component scores on new data can be computed with simple matrix multiplication. We derive explicit solutions for data matrices of special structure and provide computationally efficient algorithms for solving for the principal component loadings. Through simulation experiments and an analysis of medical diagnoses data, we compare our formulation of logistic PCA to the previous formulation as well as ordinary PCA to demonstrate its benefits.