  In high-dimensional data, structured noise caused by observed and unobserved
factors affecting multiple target variables simultaneously, imposes a serious
challenge for modeling, by masking the often weak signal. Therefore, (1)
explaining away the structured noise in multiple-output regression is of
paramount importance. Additionally, (2) assumptions about the correlation
structure of the regression weights are needed. We note that both can be
formulated in a natural way in a latent variable model, in which both the
interesting signal and the noise are mediated through the same latent factors.
Under this assumption, the signal model then borrows strength from the noise
model by encouraging similar effects on correlated targets. We introduce a
hyperparameter for the \emph{latent signal-to-noise ratio} which turns out to
be important for modelling weak signals, and an ordered infinite-dimensional
shrinkage prior that resolves the rotational unidentifiability in reduced-rank
regression models. Simulations and prediction experiments with metabolite, gene
expression, FMRI measurement, and macroeconomic time series data show that our
model equals or exceeds the state-of-the-art performance and, in particular,
outperforms the standard approach of assuming independent noise and signal
models.
