Beyond Perturbation Stability: LP Recovery Guarantees for MAP Inference on Noisy Stable Instances
This provides a theoretical explanation for the practical performance of LP relaxations in computer vision, though it is incremental as it extends existing stability results to noisy conditions.
The paper tackles the problem of MAP inference in Potts models by showing that a linear programming relaxation approximately recovers solutions even when stable instances are corrupted by noise, and demonstrates that real-world instances from computer vision have nearby stable instances.
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.