Spanning tree approximations for conditional random fields

Abstract In this work we show that one can train Conditional Random Fields of intractable graphs effectively and efficiently by considering a mixture of random spanning trees of the underlying graphical model. Furthermore, we show how a maximum-likelihood estimator of such a training objective can subsequently be used for prediction on the full graph. We present experimental results which improve on the state-of-the-art. Additionally, the training objective is less sensitive to the regularization than pseudo-likelihood based training approaches. We perform the experimental validation on two classes of data sets where structure is important: image denoising and multilabel classification

Random curves on surfaces induced from the Laplacian determinant

We define natural probability measures on cycle-rooted spanning forests (CRSFs) on graphs embedded on a surface with a Riemannian metric. These measures arise from the Laplacian determinant and depend on the choice of a unitary connection on the tangent bundle to the surface. We show that, for a sequence of graphs $(G_n)$ conformally approximating the surface, the measures on CRSFs of $G_n$ converge and give a limiting probability measure on finite multicurves (finite collections of pairwise disjoint simple closed curves) on the surface, independent of the approximating sequence. Wilson's algorithm for generating spanning trees on a graph generalizes to a cycle-popping algorithm for generating CRSFs for a general family of weights on the cycles. We use this to sample the above measures. The sampling algorithm, which relates these measures to the loop-erased random walk, is also used to prove tightness of the sequence of measures, a key step in the proof of their convergence. We set the framework for the study of these probability measures and their scaling limits and state some of their properties