So far ...

1. So far ... Exact methods for submodular energies Approximations for non-submodular energies Move-making ( N_Variables >> N_Labels)

3. Inference for LearningLinear Programming Relaxation

4. Linear Integer Programming minxg0Tx Linear function s.t. giTx ≤ 0 Linear constraints hiTx = 0 Linear constraints x is a vector of integers For example, x {0,1}N Hard to solve !!

5. Linear Programming minxg0Tx Linear function s.t. giTx ≤ 0 Linear constraints hiTx = 0 Linear constraints x is a vector of reals For example, x [0,1]N Easy to solve!! Relaxation

6. Roadmap Express MAP as an integer program Relax to a linear program and solve Round fractional solution to integers

7. Cost of V1 = 0 Cost of V1 = 1 Integer Programming Formulation 2 0 4 Unary Cost Label ‘1’ 1 3 Label ‘0’ 5 0 2 V2 V1 ; 2 4 ] 2 Unary Cost Vector u = [ 5

8. V1= 1 V1 0 Integer Programming Formulation 2 0 4 Unary Cost Label ‘1’ 1 3 Label ‘0’ 5 0 2 V2 V1 ; 2 4 ]T 2 Unary Cost Vector u = [ 5 Label vector x = [ 0 1 ; 1 0 ]T

9. Integer Programming Formulation 2 0 4 Unary Cost Label ‘1’ 1 3 Label ‘0’ 5 0 2 V2 V1 ; 2 4 ]T 2 Unary Cost Vector u = [ 5 Label vector x = [ 0 1 ; 1 0 ]T ∑iuixi Sum of Unary Costs =

10. Pairwise Cost Matrix P Cost of V1 = 0 and V1 = 0 0 Cost of V1 = 0 and V2 = 0 0 0 1 0 Cost of V1 = 0 and V2 = 1 0 1 0 0 3 0 0 0 Integer Programming Formulation 2 0 4 Pairwise Cost Label ‘1’ 1 3 Label ‘0’ 5 0 2 V2 V1 0 3 0

11. Pairwise Cost Matrix P 0 0 0 1 0 0 1 0 0 3 0 0 0 Integer Programming Formulation 2 0 4 Pairwise Cost Label ‘1’ 1 3 Label ‘0’ 5 0 2 V2 V1 Sum of Pairwise Costs ∑i<jPijxixj 0 3 0 = ∑i<jPijXij X= xxT

12. Uniqueness Constraint ∑ xi = 1 i Va Integer Programming Formulation Constraints • Integer Constraints xi{0,1} X = x xT

13. ∑ xi = 1 i  Va Integer Programming Formulation + ∑ PijXij ∑ uixi x* = argmin xi{0,1} X = x xT

14. Roadmap Express MAP as an integer program Relax to a linear program and solve Round fractional solution to integers

15. ∑ xi = 1 i  Va Non-Convex Integer Programming Formulation + ∑ PijXij ∑ uixi x* = argmin Convex xi {0,1} X = x xT

16. ∑ xi = 1 i  Va Non-Convex Integer Programming Formulation + ∑ PijXij ∑ uixi x* = argmin Convex xi [0,1] X = x xT

17. ∑ xi = 1 i  Va ∑ Xij = xi j  Vb Integer Programming Formulation + ∑ PijXij ∑ uixi x* = argmin Convex xi [0,1] Xij [0,1]

18. ∑ xi = 1 i  Va ∑ Xij = xi j  Vb Linear Programming Formulation Schlesinger, 76; Chekuri et al., 01; Wainwright et al. , 01 + ∑ PijXij ∑ uixi x* = argmin Convex xi [0,1] Xij [0,1]

19. Roadmap Express MAP as an integer program Relax to a linear program and solve Round fractional solution to integers

20. Properties Dominate many convex relaxations Kumar, Kolmogorov and Torr, 2007 Best known multiplicative bounds 2 for Potts (uniform) energies 2 + √2 for Truncated linear energies O(log n) for metric labeling Matched by move-making Kumar and Torr, 2008; Kumar and Koller, UAI 2009

21. Algorithms Tree-reweighted message passing (TRW) Max-product linear programming (MPLP) Dual decomposition Komodakis and Paragios, ICCV 2007