Log-Linear Models Local Structure Representation Probabilistic Graphical Models Local Structure Log-Linear Models

Log-Linear Representation Each feature fj has a scope Dj Different features can have same scope

Representing Table Factors (X1, X2) =

Features for Language Features: word capitalized, word in atlas or name list, previous word is “Mrs”, next word is “Times”, … Phrases

Ising Model

Metric MRFs All Xi take values in label space V Distance function  : V  V  R Reflexivity: (v,v) = 0 for all v Symmetry: (v1,v2) = (v2,v1) for all v1, v2 Triangle inequality: (v1,v2)  (v1,v3) + (v3, v2) for all v1, v2, v3 want Xi and Xj to take “similar” values Xi Xj

Metric MRFs All Xi take values in label space V Distance function  : V  V  R want Xi and Xj to take “similar” values Xi Xj values of Xi and Xj far in  lower probability

Metric MRF Examples 1 (vk,vl) = vk=vl otherwise (vk,vl) vk-vl 1 1 (vk,vl) = vk=vl otherwise (vk,vl) vk-vl (vk,vl) vk-vl

Metric MRF: Segmentation 1 1 (vk,vl) = vk=vl otherwise

Metric MRF: Denoising (vk,vl) = |vk-vl| vk-vl vk-vl (vk,vl) = max(|vk-vl|,d) Similar idea for stereo reconstruction