Tighter Relaxations for MAP-MRF Inference: A Local Primal-Dual Gap based Separation AlgorithmDhruv Batra (TTI-Chicago), Sebastian Nowozin (Microsoft Research Cambridge), Pushmeet Kohli (Microsoft Research Cambridge)

LP-Relaxations for MAP Inference in MRFs Tighter LPs and Cluster Pursuit

Graph Structure

MAP Inference LP Relaxations

[Wainwright et al. ‘08, Sontag et al. ‘07]

Reparameterization

Cluster Pursuit

What’s a good cluster score?

[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

Local Primal-Dual Gap

Complimentary Slackness Conditions

Local Primal-Dual Gap

Properties -- Positive -- Sums to current Primal-Dual Gap -- Slackness property

Results

Markov Random Fields

Variables Factors / Cliques

Energy / Cost Function

Pairwise MRF

Primal LP Dual LP

NormalizationNormalization

MarginalizationMarginalization

LagrangianLagrangian

MultipliersMultipliers

Controls Tightness of LPControls Tightness of LP

Original Factor Incoming Messages Outgoing Messages

Primal Dual

Original Image Noisy Blurry Image Pairwise LP Soln Triplet LP Soln

Dual vs. Iterations Dual vs. Time Primal-Dual Gap vs. Time

-- Synthetic experiments; Stereo; Image De-convolulation