Deep Gaussian Markov Random Fields
This work addresses spatial modeling challenges in fields like statistics, offering a novel integration of probabilistic and deep learning methods, though it is incremental in combining existing tools.
The paper tackles the problem of modeling spatial dependencies by establishing a formal connection between Gaussian Markov random fields (GMRFs) and convolutional neural networks (CNNs), generalizing GMRFs to multi-layer architectures for improved computational scaling. It demonstrates that this deep GMRF model outperforms state-of-the-art methods on a satellite temperature dataset in terms of prediction and predictive uncertainty.
Gaussian Markov random fields (GMRFs) are probabilistic graphical models widely used in spatial statistics and related fields to model dependencies over spatial structures. We establish a formal connection between GMRFs and convolutional neural networks (CNNs). Common GMRFs are special cases of a generative model where the inverse mapping from data to latent variables is given by a 1-layer linear CNN. This connection allows us to generalize GMRFs to multi-layer CNN architectures, effectively increasing the order of the corresponding GMRF in a way which has favorable computational scaling. We describe how well-established tools, such as autodiff and variational inference, can be used for simple and efficient inference and learning of the deep GMRF. We demonstrate the flexibility of the proposed model and show that it outperforms the state-of-the-art on a dataset of satellite temperatures, in terms of prediction and predictive uncertainty.