We introduce a variational Bayesian neural network where the parameters are governed via a probability distribution on random matrices. Specifically, we employ a matrix variate Gaussian \cite{gupta1999matrix} parameter posterior distribution where we explicitly model the covariance among the input and output dimensions of each layer. Furthermore, with approximate covariance matrices we can achieve a more efficient way to represent those correlations that is also cheaper than fully factorized parameter posteriors. We further show that with the "local reprarametrization trick" \cite{kingma2015variational} on this posterior distribution we arrive at a Gaussian Process \cite{rasmussen2006gaussian} interpretation of the hidden units in each layer and we, similarly with \cite{gal2015dropout}, provide connections with deep Gaussian processes. We continue in taking advantage of this duality and incorporate "pseudo-data" \cite{snelson2005sparse} in our model, which in turn allows for more efficient sampling while maintaining the properties of the original model. The validity of the proposed approach is verified through extensive experiments.
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