Efficient information planning in Gaussian MRFs

We are interested in information planning of structures represented by sparse graphical models where measurements correspond to a limited number of nodes. Choosing a set of measurements, which better describe spatiotemporal phenomena is a fundamental task whose optimal solution becomes intractable as the number of measurements grows. Krause et al. (2005) and Williams et al. (2007) have shown that by exploiting the submodular property of mutual information, a simple polynomial greedy selection algorithm comes with near-optimal guarantees. Most previous works assume oracle value models, where the value of a set of measurements is provided in constant time. However, the complexity of evaluating the reward of different measurement sets might be nontrivial in realistic settings. Here, we show that by taking advantage of sparsity in the measurement process, the complexity of information planning in Gaussian models is dramatically reduced. We additionally demonstrate that working with the information form reduces the computational load to the absolutely necessary computations. Lastly, we present an analysis of the computational complexity of different orders of selecting measurements known as visit walks, and suggest how this could help in forming a measurement schedule. We restrict ourselves to Gaussian Hidden Markov Models (HMMs), but the underlying analysis generalizes to general Markov Random Fields (MRFs).