  Several works have shown that perturbation stable instances of the MAP
inference problem in Potts models can be solved exactly using a natural linear
programming (LP) relaxation. However, most of these works give few (or no)
guarantees for the LP solutions on instances that do not satisfy the relatively
strict perturbation stability definitions. In this work, we go beyond these
stability results by showing that the LP approximately recovers the MAP
solution of a stable instance even after the instance is corrupted by noise.
This "noisy stable" model realistically fits with practical MAP inference
problems: we design an algorithm for finding "close" stable instances, and show
that several real-world instances from computer vision have nearby instances
that are perturbation stable. These results suggest a new theoretical
explanation for the excellent performance of this LP relaxation in practice.
