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

Multiple Output Regression with Latent Noise