  In unsupervised learning, dimensionality reduction is an important tool for
data exploration and visualization. Because these aims are typically
open-ended, it can be useful to frame the problem as looking for patterns that
are enriched in one dataset relative to another. These pairs of datasets occur
commonly, for instance a population of interest vs. control or signal vs.
signal free recordings.However, there are few methods that work on sets of data
as opposed to data points or sequences. Here, we present a probabilistic model
for dimensionality reduction to discover signal that is enriched in the target
dataset relative to the background dataset. The data in these sets do not need
to be paired or grouped beyond set membership. By using a probabilistic model
where some structure is shared amongst the two datasets and some is unique to
the target dataset, we are able to recover interesting structure in the latent
space of the target dataset. The method also has the advantages of a
probabilistic model, namely that it allows for the incorporation of prior
information, handles missing data, and can be generalized to different
distributional assumptions. We describe several possible variations of the
model and demonstrate the application of the technique to de-noising, feature
selection, and subgroup discovery settings.
