Efficient methods for Gaussian Markov random fields under sparse linear constraints
This addresses computational bottlenecks for researchers and practitioners using constrained GMRFs in applications like Gaussian process regression, though it appears incremental as an optimization of existing methods.
The paper tackles the computational challenges of inference and simulation for linearly constrained Gaussian Markov Random Fields (GMRFs) when constraints are numerous, proposing a new class of methods for sparse constraints that greatly outperforms existing alternatives in computational cost, as demonstrated in applications with simulated data.
Methods for inference and simulation of linearly constrained Gaussian Markov Random Fields (GMRF) are computationally prohibitive when the number of constraints is large. In some cases, such as for intrinsic GMRFs, they may even be unfeasible. We propose a new class of methods to overcome these challenges in the common case of sparse constraints, where one has a large number of constraints and each only involves a few elements. Our methods rely on a basis transformation into blocks of constrained versus non-constrained subspaces, and we show that the methods greatly outperform existing alternatives in terms of computational cost. By combining the proposed methods with the stochastic partial differential equation approach for Gaussian random fields, we also show how to formulate Gaussian process regression with linear constraints in a GMRF setting to reduce computational cost. This is illustrated in two applications with simulated data.