  Non-Negative Matrix Factorization (NMF) is a widely used dimension reduction
method that factorizes a non-negative data matrix into two lower dimensional
non-negative matrices: One is the basis or feature matrix which consists of the
variables and the other is the coefficients matrix which is the projections of
data points to the new basis. The features can be interpreted as sub-structures
of the data. The number of sub-structures in the feature matrix is also called
the rank which is the only tuning parameter in NMF. An appropriate rank will
extract the key latent features while minimizing the noise from the original
data. In this paper, we develop a novel rank selection method based on
hypothesis testing, using a deconvolved bootstrap distribution to assess the
significance level accurately despite the large amount of optimization error.
In the simulation section, we compare our method with a rank selection method
based on hypothesis testing using bootstrap distribution without deconvolution,
and with a cross-validated imputation method1. Through simulations, we
demonstrate that our method is not only accurate at estimating the true ranks
for NMF especially when the features are hard to distinguish but also efficient
at computation. When applied to real microbiome data (e.g. OTU data and
functional metagenomic data), our method also shows the ability to extract
interpretable sub-communities in the data.
