  This paper proposes using a sparse-structured multivariate Gaussian to
provide a closed-form approximator for the output of probabilistic ensemble
models used for dense image prediction tasks. This is achieved through a
convolutional neural network that predicts the mean and covariance of the
distribution, where the inverse covariance is parameterised by a sparsely
structured Cholesky matrix. Similarly to distillation approaches, our single
network is trained to maximise the probability of samples from pre-trained
probabilistic models, in this work we use a fixed ensemble of networks. Once
trained, our compact representation can be used to efficiently draw spatially
correlated samples from the approximated output distribution. Importantly, this
approach captures the uncertainty and structured correlations in the
predictions explicitly in a formal distribution, rather than implicitly through
sampling alone. This allows direct introspection of the model, enabling
visualisation of the learned structure. Moreover, this formulation provides two
further benefits: estimation of a sample probability, and the introduction of
arbitrary spatial conditioning at test time. We demonstrate the merits of our
approach on monocular depth estimation and show that the advantages of our
approach are obtained with comparable quantitative performance.
