  Two key challenges in modern statistical applications are the large amount of
information recorded per individual, and that such data are often not collected
all at once but in batches. These batch effects can be complex, causing
distortions in both mean and variance. We propose a novel sparse latent factor
regression model to integrate such heterogeneous data. The model provides a
tool for data exploration via dimensionality reduction while correcting for a
range of batch effects. We study the use of several sparse priors (local and
non-local) to learn the dimension of the latent factors. Our model is fitted in
a deterministic fashion by means of an EM algorithm for which we derive
closed-form updates, contributing a novel scalable algorithm for non-local
priors of interest beyond the immediate scope of this paper. We present several
examples, with a focus on bioinformatics applications. Our results show an
increase in the accuracy of the dimensionality reduction, with non-local priors
substantially improving the reconstruction of factor cardinality, as well as
the need to account for batch effects to obtain reliable results. Our model
provides a novel approach to latent factor regression that balances sparsity
with sensitivity and is highly computationally efficient.
