  Sparse latent multi-factor models have been used in many exploratory and
predictive problems with high-dimensional multivariate observations. Because of
concerns with identifiability, the latent factors are almost always assumed to
be linearly related to measured feature variables. Here we explore the analysis
of multi-factor models with different structures of interactions between latent
factors, including multiplicative effects as well as a more general framework
for nonlinear interactions introduced via the Gaussian Process. We utilize
sparsity priors to test whether the factors and interaction terms have
significant effect. The performance of the models is evaluated through
simulated and real data applications in genomics. Variation in the number of
copies of regions of the genome is a well-known and important feature of most
cancers. We examine interactions between factors directly associated with
different chromosomal regions detected with copy number alteration in breast
cancer data. In this context, significant interaction effects for specific
genes suggest synergies between duplications and deletions in different regions
of the chromosome.
