  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.
