Local Rules for Global MAP: When Do They Work ?

We consider the question of computing Maximum A Posteriori (MAP) assignmentin an arbitrary pair-wise Markov Random Field (MRF). We present a randomizediterative algorithm based on simple local updates. The algorithm, starting with anarbitrary initial assignment, updates it in each iteration by first, picking a randomnode, then selecting an (appropriately chosen) random local neighborhood andoptimizing over this local neighborhood. Somewhat surprisingly, we show thatthis algorithm finds a near optimal assignment within n log n iterations with highprobability for any n node pair-wise MRF with geometry (i.e. MRF graph withpolynomial growth) with the approximation error depending on (in a reasonablemanner) the geometric growth rate of the graph and the average radius of the localneighborhood – this allows for a graceful tradeoff between the complexity of thealgorithm and the approximation error. Through extensive simulations, we showthat our algorithm finds extremely good approximate solutions for various kindsof MRFs with geometry.