Beyond Perturbation Stability: LP Recovery Guarantees for MAP Inference on Noisy Stable Instances

TL;DR

This paper extends the understanding of LP relaxation in MAP inference by demonstrating its robustness to noise in stable instances, supported by algorithms and real-world data analysis.

Contribution

It introduces a noisy stable model for MAP inference, showing LP can approximately recover solutions even with noisy data, and provides algorithms to find stable instances close to real-world problems.

Findings

LP relaxation approximately recovers MAP solutions in noisy stable instances

Real-world computer vision instances have nearby stable instances

The results explain LP relaxation's practical effectiveness

Abstract

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.