Ising Graphical Model
Abstract
The Ising model is an important model in statistical physics, with over 10,000 papers published on the topic. This model assumes binary variables and only local pairwise interactions between neighbouring nodes. Inference for the general Ising model is NP-hard; this includes tasks such as calculating the partition function, finding a lowest-energy (ground) state and computing marginal probabilities. Past approaches have proceeded by working with classes of tractable Ising models, such as Ising models defined on a planar graph. For such models, the partition function and ground state can be computed exactly in polynomial time by establishing a correspondence with perfect matchings in a related graph. In this thesis we continue this line of research. In particular we simplify previous inference algorithms for the planar Ising model. The key to our construction is the complementary correspondence between graph cuts of the model graph and perfect matchings of its expanded dual. We show that our exact algorithms are effective and efficient on a number of real-world machine learning problems. We also investigate heuristic methods for approximating ground states of non-planar Ising models. We show that in this setting our approximative algorithms are superior than current state-of-the-art methods.