Tighter Relaxations for MAP-MRF Inference: A Local Primal-Dual Gap based Separation Algorithm

1. Tighter Relaxations for MAP-MRF Inference: A Local Primal-Dual Gap based Separation Algorithm Dhruv Batra (TTI-Chicago), Sebastian Nowozin (Microsoft Research Cambridge), Pushmeet Kohli (Microsoft Research Cambridge) Local Primal-Dual Gap LP-Relaxations for MAP Inference in MRFs Tighter LPs and Cluster Pursuit Markov Random Fields Primal LP Dual LP Complimentary Slackness Conditions Graph Structure Normalization Lagrangian Local Primal-Dual Gap Marginalization Multipliers Variables Factors / Cliques Primal Dual Properties -- Positive -- Sums to current Primal-Dual Gap -- Slackness property Controls Tightness of LP Reparameterization Results Energy / Cost Function Pairwise MRF -- Synthetic experiments; Stereo; Image De-convolulation Original Factor Incoming Messages Outgoing Messages Cluster Pursuit Original Image Noisy Blurry Image Pairwise LP Soln Triplet LP Soln MAP Inference LP Relaxations What’s a good cluster score? [Wainwright et al. ‘08, Sontag et al. ‘07] Dual vs. Iterations Dual vs. Time Primal-Dual Gap vs. Time [Sontag et al. UAI ’08] Lower-bound on improvement in Dual [Werner CVPR ’08] Try each cluster and check improvement [Komodakis et al. ECCV ‘08] PROPOSED: A surrogate score -- Efficiently computable -- Correlated with increase in Dual -- Motivated by LP duality Complimentary Slackness