1. MAP Inference in Discrete Models
M. Pawan Kumar, Stanford University

2. The Problem
E(x) = ∑ fi (xi) + ∑ gij (xi,xj) + ∑ hc(xc)
i ij c
Unary Pairwise Higher Order
Problems worthy of attack
Prove their worth by fighting back
Minimize E(x) ….. Done !!!

3. Energy Minimization Problems
NP-Hard
MAXCUT
CSP NP-hard
Space of Problems

4. The Issues
• Which functions are exactly solvable?
Boros Hammer [1965], Kolmogorov Zabih [ECCV 2002, PAMI 2004] , Ishikawa [PAMI
2003], Schlesinger [EMMCVPR 2007], Kohli Kumar Torr [CVPR2007, PAMI 2008] ,
Ramalingam Kohli Alahari Torr [CVPR 2008] , Kohli Ladicky Torr [CVPR 2008, IJCV
2009] , Zivny Jeavons [CP 2008]
• Approximate solutions of NP-hard problems
Schlesinger [76 ], Kleinberg and Tardos [FOCS 99], Chekuri et al. [01], Boykov et al.
[PAMI 01], Wainwright et al. [NIPS01], Werner [PAMI 2007], Komodakis et al. [PAMI, 05
07], Lempitsky et al. [ICCV 2007], Kumar et al. [NIPS 2007], Kumar et al. [ICML 2008],
Sontag and Jakkola [NIPS 2007], Kohli et al. [ICML 2008], Kohli et al. [CVPR 2008,
IJCV 2009], Rother et al. [2009]
• Scalability and Efficiency
Kohli Torr [ICCV 2005, PAMI 2007], Juan and Boykov [CVPR 2006], Alahari Kohli Torr
[CVPR 2008] , Delong and Boykov [CVPR 2008]

5. The Issues
• Which functions are exactly solvable?
Boros Hammer [1965], Kolmogorov Zabih [ECCV 2002, PAMI 2004] , Ishikawa [PAMI
2003], Schlesinger [EMMCVPR 2007], Kohli Kumar Torr [CVPR2007, PAMI 2008] ,
Ramalingam Kohli Alahari Torr [CVPR 2008] , Kohli Ladicky Torr [CVPR 2008, IJCV
2009] , Zivny Jeavons [CP 2008]
• Approximate solutions of NP-hard problems
Schlesinger [76 ], Kleinberg and Tardos [FOCS 99], Chekuri et al. [01], Boykov et al.
[PAMI 01], Wainwright et al. [NIPS01], Werner [PAMI 2007], Komodakis et al. [PAMI, 05
07], Lempitsky et al. [ICCV 2007], Kumar et al. [NIPS 2007], Kumar et al. [ICML 2008],
Sontag and Jakkola [NIPS 2007], Kohli et al. [ICML 2008], Kohli et al. [CVPR 2008,
IJCV 2009], Rother et al. [2009]
• Scalability and Efficiency
Kohli Torr [ICCV 2005, PAMI 2007], Juan and Boykov [CVPR 2006], Alahari Kohli Torr
[CVPR 2008] , Delong and Boykov [CVPR 2008]

6. Popular Inference Methods
 Iterated Conditional Modes (ICM)
Classical Move making
 Simulated Annealing algorithms
 Dynamic Programming (DP)
 Belief Propagtion (BP) Message passing
 Tree-Reweighted (TRW), Diffusion
 Graph Cut (GC)
Combinatorial Algorithms
 Branch & Bound
 Relaxation methods: Convex Optimization
(Linear Programming,
...)
 …

7. Road Map
9.30-10.00 Introduction (Andrew Blake)
10.00-11.00 Discrete Models in Computer Vision (Carsten Rother)
15min Coffee break
11.15-12.30 Message Passing: DP, TRW, LP relaxation (Pawan
Kumar)
12.30-13.00 Quadratic pseudo-boolean optimization (Pushmeet
Kohli)
1hour Lunch break
14:00-15.00 Transformation and move-making methods (Pushmeet
Kohli)
15:00-15.30 Speed and Efficiency (Pushmeet Kohli)
15min Coffee break
15:45-16.15 Comparison of Methods (Carsten Rother)
16:30-17.30 Recent Advances: Dual-decomposition, higher-order,
etc. (Carsten Rother + Pawan Kumar)

8. Notation
Random variables - Va, Vb, ….
Labels - li, lj, ….
Labeling - f : {a,b,..} {i,j, …}
Unary Potential - a;f(a)
Pairwise Potential - ab;f(a)f(b)

9. Notation
Energy Function
Q(f; ) = ∑a a;f(a) + ∑(a,b) ab;f(a)f(b)
MAP Estimation
f* = arg min Q(f; )
Min-marginals
qa;i = min Q(f; ) s.t. f(a) = i

10. Outline
• Reparameterization
• Belief Propagation
• Tree-reweighted Message Passing

11. Reparameterization
2+2 0 4 -2 f(a) f(b) Q(f; )
1 1 0 0 7
0 1 10
2+ 5 0 2 -2 1 0 5
Va Vb
1 1 6
Add a constant to all a;i
Subtract that constant from all b;k

12. Reparameterization
2+2 0 4 -2 f(a) f(b) Q(f; )
1 1 0 0 7 +2-2
0 1 10 + 2 - 2
2+ 5 0 2 -2 1 0 5 +2-2
Va Vb
1 1 6 +2-2
Add a constant to all a;i
Subtract that constant from all b;k
Q(f; ’) = Q(f; )

13. Reparameterization
2 0-3 4 +3 f(a) f(b) Q(f; )
1 1- 3 0 0 7
0 1 10
5 0 2 1 0 5
Va Vb
1 1 6
Add a constant to one b;k
Subtract that constant from ab;ik for all ‘i’

14. Reparameterization
2 0-3 4 +3 f(a) f(b) Q(f; )
1 1- 3 0 0 7
0 1 10 - 3 + 3
5 0 2 1 0 5
Va Vb
1 1 6-3+3
Add a constant to one b;k
Subtract that constant from ab;ik for all ‘i’
Q(f; ’) = Q(f; )

15. Reparameterization
3 1 3 1 3 0-4 1+ 4
1+ 1
2 2-2 4 2 4 -1 2 1- 4 4
0+ 1
1- 2 2-4
1+ 1
5 0-2 2 +2 5 2 5 2
Va Vb Va Vb Va Vb
’a;i = a;i + Mba;i ’b;k = b;k + Mab;k
Q(f; ’)
’ab;ik = ab;ik - Mab;k - Mba;i
= Q(f; )

16. Reparameterization
’ is a reparameterization of , iff
Q(f; ’) = Q(f; ), for all f ’
Equivalently Kolmogorov, PAMI, 2006
2+2 0 4 -2
’a;i = a;i + Mba;i
1 1
’b;k = b;k + Mab;k
2+ 5 0 2 -2
Va Vb
’ab;ik = ab;ik - Mab;k - Mba;i

17. Recap
MAP Estimation
f* = arg min Q(f; )
Q(f; ) = ∑a a;f(a) + ∑(a,b) ab;f(a)f(b)
Min-marginals
qa;i = min Q(f; ) s.t. f(a) = i
Reparameterization
Q(f; ’) = Q(f; ), for all f ’

18. Outline
• Reparameterization
• Belief Propagation
– Exact MAP for Chains and Trees
– Approximate MAP for general graphs
– Computational Issues and Theoretical Properties
• Tree-reweighted Message Passing

19. Belief Propagation
• Some MAP problems are easy
• Belief Propagation gives exact MAP for chains
• Exact MAP for trees
• Clever Reparameterization

20. Two Variables
2 2 0 4
1 1
5 0 2 5
Va Vb Va Vb
Add a constant to one b;k
Subtract that constant from ab;ik for all ‘i’
Choose the right constant ’b;k = qb;k

21. Two Variables
2 2 0 4
1 1
5 0 2 5
Va Vb Va Vb
a;0 + ab;00 = 5+0
Mab;0 = min
a;1 + ab;10 = 2+1
Choose the right constant ’b;k = qb;k

22. Two Variables
2 2 0 4
-2 1
5 -3 5 5
Va Vb Va Vb
Choose the right constant ’b;k = qb;k

23. Two Variables
2
f(a) = 1 2 0 4
-2 1
5 -3 5 5
Va Vb Va Vb
’b;0 = qb;0
Potentials along the red path add up to 0
Choose the right constant ’b;k = qb;k

24. Two Variables
2 2 0 4
-2 1
5 -3 5 5
Va Vb Va Vb
a;0 + ab;01 = 5+1
Mab;1 = min
a;1 + ab;11 = 2+0
Choose the right constant ’b;k = qb;k

25. Two Variables
f(a) = 1 f(a) = 1
2 2 -2 6
-2 -1
5 -3 5 5
Va Vb Va Vb
’b;0 = qb;0 ’b;1 = qb;1
Minimum of min-marginals = MAP estimate
Choose the right constant ’b;k = qb;k

26. Two Variables
f(a) = 1 f(a) = 1
2 2 -2 6
-2 -1
5 -3 5 5
Va Vb Va Vb
’b;0 = qb;0 ’b;1 = qb;1
f*(b) = 0 f*(a) = 1
Choose the right constant ’b;k = qb;k

27. Two Variables
f(a) = 1 f(a) = 1
2 2 -2 6
-2 -1
5 -3 5 5
Va Vb Va Vb
’b;0 = qb;0 ’b;1 = qb;1
We get all the min-marginals of Vb
Choose the right constant ’b;k = qb;k

28. Recap
We only need to know two sets of equations
General form of Reparameterization
’a;i = a;i + Mba;i ’b;k = b;k+ Mab;k
’ab;ik = ab;ik- Mab;k- Mba;i
Reparameterization of (a,b) in Belief Propagation
Mab;k = mini { a;i + ab;ik }
Mba;i = 0

29. Three Variables
l1 2 0 4 0 6
1 1 2 3
l0
5 0 2 1 3
Va Vb Vc
Reparameterize the edge (a,b) as before

30. Three Variables
f(a) = 1
l1 2 -2 6 0 6
-2 -1 2 3
l0
5 -3 5 1 3
Va Vb Vc
f(a) = 1
Reparameterize the edge (a,b) as before

31. Three Variables
f(a) = 1
l1 2 -2 6 0 6
-2 -1 2 3
l0
5 -3 5 1 3
Va Vb Vc
f(a) = 1
Reparameterize the edge (a,b) as before
Potentials along the red path add up to 0

32. Three Variables
f(a) = 1
l1 2 -2 6 0 6
-2 -1 2 3
l0
5 -3 5 1 3
Va Vb Vc
f(a) = 1
Reparameterize the edge (b,c) as before
Potentials along the red path add up to 0

33. Three Variables
f(a) = 1 f(b) = 1
l1 2 -2 6 -6 12
-2 -1 -4 -3
l0
5 -3 5 -5 9
Va Vb Vc
f(a) = 1 f(b) = 0
Reparameterize the edge (b,c) as before
Potentials along the red path add up to 0

34. Three Variables
f(a) = 1 f(b) = 1
l1 2 -2 6 -6 12
qc;1
-2 -1 -4 -3
l0 qc;0
5 -3 5 -5 9
Va Vb Vc
f(a) = 1 f(b) = 0
Reparameterize the edge (b,c) as before
Potentials along the red path add up to 0

35. Three Variables
f(a) = 1 f(b) = 1
l1 2 -2 6 -6 12
qc;1
-2 -1 -4 -3
l0 qc;0
5 -3 5 -5 9
Va Vb Vc
f(a) = 1 f(b) = 0
f*(c) = 0 f*(b) = 0 f*(a) = 1
Generalizes to any length chain

36. Three Variables
f(a) = 1 f(b) = 1
l1 2 -2 6 -6 12
qc;1
-2 -1 -4 -3
l0 qc;0
5 -3 5 -5 9
Va Vb Vc
f(a) = 1 f(b) = 0
f*(c) = 0 f*(b) = 0 f*(a) = 1
Only Dynamic Programming

37. Why Dynamic Programming?
3 variables 2 variables + book-keeping
n variables (n-1) variables + book-keeping
Start from left, go to right
Reparameterize current edge (a,b)
Mab;k = mini { a;i + ab;ik }
’b;k = b;k+ Mab;k ’ab;ik = ab;ik- Mab;k
Repeat

38. Why Dynamic Programming?
Messages Message Passing
Why stop at dynamic programming?
Start from left, go to right
Reparameterize current edge (a,b)
Mab;k = mini { a;i + ab;ik }
’b;k = b;k+ Mab;k ’ab;ik = ab;ik- Mab;k
Repeat

39. Three Variables
l1 2 -2 6 -6 12
-2 -1 -4 -3
l0
5 -3 5 -5 9
Va Vb Vc
Reparameterize the edge (c,b) as before

40. Three Variables
l1 2 -2 11 -11 12
-2 -1 -9 -7
l0
5 -3 9 -9 9
Va Vb Vc
Reparameterize the edge (c,b) as before
’b;i = qb;i

41. Three Variables
l1 2 -2 11 -11 12
-2 -1 -9 -7
l0
5 -3 9 -9 9
Va Vb Vc
Reparameterize the edge (b,a) as before

42. Three Variables
l1 9 -9 11 -11 12
-9 -7 -9 -7
l0
11 -9 9 -9 9
Va Vb Vc
Reparameterize the edge (b,a) as before
’a;i = qa;i

43. Three Variables
l1 9 -9 11 -11 12
-9 -7 -9 -7
l0
11 -9 9 -9 9
Va Vb Vc
Forward Pass   Backward Pass
All min-marginals are computed

44. Belief Propagation on Chains
Start from left, go to right
Reparameterize current edge (a,b)
Mab;k = mini { a;i + ab;ik }
’b;k = b;k+ Mab;k ’ab;ik = ab;ik- Mab;k
Repeat till the end of the chain
Start from right, go to left
Repeat till the end of the chain

45. Belief Propagation on Chains
• Generalizes to chains of any length
• A way of computing reparam constants
• Forward Pass - Start to End
• MAP estimate
• Min-marginals of final variable
• Backward Pass - End to start
• All other min-marginals
Won’t need this .. But good to know

46. Computational Complexity
• Each constant takes O(|L|)
• Number of constants - O(|E||L|)
O(|E||L|2)
• Memory required ?
O(|E||L|)

47. Belief Propagation on Trees
Va
Vb Vc
Vd Ve Vg Vh
Forward Pass: Leaf  Root
Backward Pass: Root  Leaf
All min-marginals are computed

48. Outline
• Reparameterization
• Belief Propagation
– Exact MAP for Chains and Trees
– Approximate MAP for general graphs
– Computational Issues and Theoretical Properties
• Tree-reweighted Message Passing

49. Belief Propagation on Cycles
a;1 b;1
a;0 b;0
Va Vb
d;1 c;1
d;0 c;0
Vd Vc
Where do we start? Arbitrarily
Reparameterize (a,b)

50. Belief Propagation on Cycles
a;1
’b;1
a;0
’b;0
Va Vb
d;1 c;1
d;0 c;0
Vd Vc
Potentials along the red path add up to 0

51. Belief Propagation on Cycles
a;1
’b;1
a;0
’b;0
Va Vb
d;1
’c;1
d;0
’c;0
Vd Vc
Potentials along the red path add up to 0

52. Belief Propagation on Cycles
a;1
’b;1
a;0
’b;0
Va Vb
’d;1 ’c;1
’d;0 ’c;0
Vd Vc
Potentials along the red path add up to 0

53. Belief Propagation on Cycles
’a;1 - a;1 ’b;1
’a;0 ’b;0
- a;0
Va Vb
’d;1 ’c;1
’d;0 ’c;0
Vd Vc
Did not obtain min-marginals
Potentials along the red path add up to 0

54. Belief Propagation on Cycles
’a;1 - a;1 ’b;1
’a;0 ’b;0
- a;0
Va Vb
’d;1 ’c;1
’d;0 ’c;0
Vd Vc
Reparameterize (a,b) again

55. Belief Propagation on Cycles
’a;1 ’’b;1
’a;0 ’’b;0
Va Vb
’d;1 ’c;1
’d;0 ’c;0
Vd Vc
Reparameterize (a,b) again
But doesn’t this overcount some potentials?

56. Belief Propagation on Cycles
’a;1 ’’b;1
’a;0 ’’b;0
Va Vb
’d;1 ’c;1
’d;0 ’c;0
Vd Vc
Reparameterize (a,b) again
Yes. But we will do it anyway

57. Belief Propagation on Cycles
’a;1 ’’b;1
’a;0 ’’b;0
Va Vb
’d;1 ’c;1
’d;0 ’c;0
Vd Vc
Keep reparameterizing edges in some order
No convergence guarantees

58. Belief Propagation
• Generalizes to any arbitrary random field
• Complexity per iteration ?
O(|E||L|2)
• Memory required ?
O(|E||L|)

59. Outline
• Reparameterization
• Belief Propagation
– Exact MAP for Chains and Trees
– Approximate MAP for general graphs
– Computational Issues and Theoretical Properties
• Tree-reweighted Message Passing

60. Computational Issues of BP
Complexity per iteration O(|E||L|2)
Special Pairwise Potentials ab;ik = wabd(|i-k|)
d d d
i-k i-k i-k
Potts Truncated Linear Truncated Quadratic
O(|E||L|) Felzenszwalb & Huttenlocher, 2004

61. Computational Issues of BP
Memory requirements O(|E||L|)
Half of original BP Kolmogorov, 2006
Some approximations exist
Yu, Lin, Super and Tan, 2007
Lasserre, Kannan and Winn, 2007
But memory still remains an issue

62. Computational Issues of BP
Order of reparameterization
Randomly
In some fixed order
The one that results in maximum change
Residual Belief Propagation
Elidan et al. , 2006

63. Summary of BP
Exact for chains
Exact for trees
Approximate MAP for general cases
Convergence not guaranteed
So can we do something better?

64. Outline
• Reparameterization
• Belief Propagation
• Tree-reweighted Message Passing
– Integer Programming Formulation
– Linear Programming Relaxation and its Dual
– Convergent Solution for Dual
– Computational Issues and Theoretical
Properties

65. Integer Programming Formulation
Unary Potentials Label l1
2 0 4
1 1 2
a;0 =5 b;0 =2
Label l0
=4 5 0 2
a;1 = 2 b;1
Va Vb
Labeling
f(a) = 1 ya;0 = 0 ya;1 = 1
f(b) = 0 yb;0 = 1 yb;1 = 0
Any f(.) has equivalent boolean variables ya;i

66. Integer Programming Formulation
Unary Potentials Label l1
2 0 4
1 1 2
a;0 =5 b;0 =2
Label l0
=4 5 0 2
a;1 = 2 b;1
Va Vb
Labeling
f(a) = 1 ya;0 = 0 ya;1 = 1
f(b) = 0 yb;0 = 1 yb;1 = 0
Find the optimal variables ya;i

67. Integer Programming Formulation
Unary Potentials Label l1
2 0 4
1 1 2
a;0 =5 b;0 =2
Label l0
=4 5 0 2
a;1 = 2 b;1
Va Vb
Sum of Unary Potentials
∑a ∑i a;i ya;i
ya;i {0,1}, for all Va, li
∑i ya;i = 1, for all Va

68. Integer Programming Formulation
Pairwise Potentials Label l1
2 0 4
1 1 2
ab;00 =0 ab;01 =1
Label l0
=0 5 0 2
ab;10 = 1 ab;11
Va Vb
Sum of Pairwise Potentials
∑(a,b) ∑ik ab;ik ya;iyb;k
ya;i {0,1}
∑i ya;i = 1

69. Integer Programming Formulation
Pairwise Potentials Label l1
2 0 4
1 1 2
ab;00 =0 ab;01 =1
Label l0
=0 5 0 2
ab;10 = 1 ab;11
Va Vb
Sum of Pairwise Potentials
∑(a,b) ∑ik ab;ik yab;ik
ya;i {0,1} yab;ik = ya;i yb;k
∑i ya;i = 1

70. Integer Programming Formulation
min ∑a ∑i a;i ya;i + ∑(a,b) ∑ik ab;ik yab;ik
ya;i {0,1}
∑i ya;i = 1
yab;ik = ya;i yb;k

71. Integer Programming Formulation
min Ty
ya;i {0,1}
∑i ya;i = 1
yab;ik = ya;i yb;k
=[… a;i …. ; … ab;ik ….]
y = [ … ya;i …. ; … yab;ik ….]

72. Integer Programming Formulation
min Ty
ya;i {0,1}
∑i ya;i = 1
yab;ik = ya;i yb;k
Solve to obtain MAP labeling y*

73. Integer Programming Formulation
min Ty
ya;i {0,1}
∑i ya;i = 1
yab;ik = ya;i yb;k
But we can’t solve it in general

74. Outline
• Reparameterization
• Belief Propagation
• Tree-reweighted Message Passing
– Integer Programming Formulation
– Linear Programming Relaxation and its Dual
– Convergent Solution for Dual
– Computational Issues and Theoretical
Properties

75. Linear Programming Relaxation
min Ty
ya;i {0,1}
∑i ya;i = 1
yab;ik = ya;i yb;k
Two reasons why we can’t solve this

76. Linear Programming Relaxation
min Ty
ya;i [0,1]
∑i ya;i = 1
yab;ik = ya;i yb;k
One reason why we can’t solve this

77. Linear Programming Relaxation
min Ty
ya;i [0,1]
∑i ya;i = 1
∑k yab;ik = ∑kya;i yb;k
One reason why we can’t solve this

78. Linear Programming Relaxation
min Ty
ya;i [0,1]
∑i ya;i = 1
∑k yab;ik = ya;i∑k yb;k =1
One reason why we can’t solve this

79. Linear Programming Relaxation
min Ty
ya;i [0,1]
∑i ya;i = 1
∑k yab;ik = ya;i
One reason why we can’t solve this

80. Linear Programming Relaxation
min Ty
ya;i [0,1]
∑i ya;i = 1
∑k yab;ik = ya;i
No reason why we can’t solve this*
*memory requirements, time complexity

81. Dual of the LP Relaxation
Wainwright et al., 2001
min Ty
Va Vb Vc
Vd Ve Vf
ya;i [0,1]
Vg Vh Vi
∑i ya;i = 1
∑k yab;ik = ya;i

82. Dual of the LP Relaxation
Wainwright et al., 2001
1 1
Va Vb Vc
Va Vb Vc
2 2
Vd Ve Vf
Vd Ve Vf
3 Vg Vh Vi 3
Vg Vh Vi
4 5 6
Va Vb Vc
i≥ 0
Vd Ve Vf
Vg Vh Vi
i i = 4 5 6

83. Dual of the LP Relaxation
Wainwright et al., 2001
1
q*( 1) Va Vb Vc
Va Vb Vc
2
q*( 2) Vd Ve Vf
Vd Ve Vf
3
q*( 3) Vg Vh Vi
Vg Vh Vi
q*( 4) q*( 5) q*( 6)
Va Vb Vc
i≥ 0
Dual of LP Vd Ve Vf
max i q*( i)
Vg Vh Vi
i i = 4 5 6

84. Dual of the LP Relaxation
Wainwright et al., 2001
1
q*( 1) Va Vb Vc
Va Vb Vc
2
q*( 2) Vd Ve Vf
Vd Ve Vf
3
q*( 3) Vg Vh Vi
Vg Vh Vi
q*( 4) q*( 5) q*( 6)
Va Vb Vc
i≥ 0
Dual of LP Vd Ve Vf
max i q*( i)
Vg Vh Vi
i i 4 5 6

85. Dual of the LP Relaxation
Wainwright et al., 2001
max i q*( i)
i i
I can easily compute q*( i)
I can easily maintain reparam constraint
So can I easily solve the dual?

86. Outline
• Reparameterization
• Belief Propagation
• Tree-reweighted Message Passing
– Integer Programming Formulation
– Linear Programming Relaxation and its Dual
– Convergent Solution for Dual
– Computational Issues and Theoretical
Properties

87. TRW Message Passing
Kolmogorov, 2006 4 5 6
1 V 1
a Vb Vc Va Vb Vc
2 2
Vd Ve Vf Vd Ve Vf
3 V Vh Vi 3
g Vg Vh Vi
4 5 6
Pick a variable Va
i q*( i)
i i

88. TRW Message Passing
Kolmogorov, 2006
1 1 1 4 4 4
c;1 b;1 a;1 a;1 d;1 g;1
1 1 1 4 4 4
c;0 b;0 a;0 a;0 d;0 g;0
Vc Vb Va Va Vd Vg
i q*( i)
i i

89. TRW Message Passing
Kolmogorov, 2006
1 1 1 4 4 4
c;1 b;1 a;1 a;1 d;1 g;1
1 1 1 4 4 4
c;0 b;0 a;0 a;0 d;0 g;0
Vc Vb Va Va Vd Vg
Reparameterize to obtain min-marginals of Va
1 q*( 1) + 4 q*( 4) +K
1 1+ 4 4 + rest

90. TRW Message Passing
Kolmogorov, 2006
’1c;1 ’1b;1 ’1a;1 ’4a;1 ’4d;1 ’4g;1
’1c;0 ’1b;0 ’1a;0 ’4a;0 ’4d;0 ’4g;0
Vc Vb Va Va Vd Vg
One pass of Belief Propagation
1 q*( ’1) + 4 q*( ’4) + K
1 ’1 + 4 ’4 + rest

91. TRW Message Passing
Kolmogorov, 2006
’1c;1 ’1b;1 ’1a;1 ’4a;1 ’4d;1 ’4g;1
’1c;0 ’1b;0 ’1a;0 ’4a;0 ’4d;0 ’4g;0
Vc Vb Va Va Vd Vg
Remain the same
1 q*( ’1) + 4 q*( ’4) + K
1 ’1 + 4 ’4 + rest

92. TRW Message Passing
Kolmogorov, 2006
’1c;1 ’1b;1 ’1a;1 ’4a;1 ’4d;1 ’4g;1
’1c;0 ’1b;0 ’1a;0 ’4a;0 ’4d;0 ’4g;0
Vc Vb Va Va Vd Vg
1 min{ ’1a;0, ’1a;1} + 4 min{ ’4a;0, ’4a;1} + K
1 ’1 + 4 ’4 + rest

93. TRW Message Passing
Kolmogorov, 2006
’1c;1 ’1b;1 ’1a;1 ’4a;1 ’4d;1 ’4g;1
’1c;0 ’1b;0 ’1a;0 ’4a;0 ’4d;0 ’4g;0
Vc Vb Va Va Vd Vg
Compute weighted average of min-marginals of Va
1 min{ ’1a;0, ’1a;1} + 4 min{ ’4a;0, ’4a;1} + K
1 ’1 + 4 ’4 + rest

94. TRW Message Passing
Kolmogorov, 2006
’1c;1 ’1b;1 ’1a;1 ’4a;1 ’4d;1 ’4g;1
’1c;0 ’1b;0 ’1a;0 ’4a;0 ’4d;0 ’4g;0
Vc Vb Va Va Vd Vg
’’a;0 = 1 ’1a;0+ 4 ’4a;0 ’’a;1 = 1 ’1a;1+ 4 ’4a;1
1 + 4 1 + 4
1 min{ ’1a;0, ’1a;1} + 4 min{ ’4a;0, ’4a;1} + K
1 ’1 + 4 ’4 + rest

95. TRW Message Passing
Kolmogorov, 2006
’1c;1 ’1b;1 ’’a;1 ’’a;1 ’4d;1 ’4g;1
’1c;0 ’1b;0 ’’a;0 ’’a;0 ’4d;0 ’4g;0
Vc Vb Va Va Vd Vg
’’a;0 = 1 ’1a;0+ 4 ’4a;0 ’’a;1 = 1 ’1a;1+ 4 ’4a;1
1 + 4 1 + 4
1 min{ ’1a;0, ’1a;1} + 4 min{ ’4a;0, ’4a;1} + K
1 ’’1 + 4 ’’4 + rest

96. TRW Message Passing
Kolmogorov, 2006
’1c;1 ’1b;1 ’’a;1 ’’a;1 ’4d;1 ’4g;1
’1c;0 ’1b;0 ’’a;0 ’’a;0 ’4d;0 ’4g;0
Vc Vb Va Va Vd Vg
’’a;0 = 1 ’1a;0+ 4 ’4a;0 ’’a;1 = 1 ’1a;1+ 4 ’4a;1
1 + 4 1 + 4
1 min{ ’1a;0, ’1a;1} + 4 min{ ’4a;0, ’4a;1} + K
1 ’’1 + 4 ’’4 + rest

97. TRW Message Passing
Kolmogorov, 2006
’1c;1 ’1b;1 ’’a;1 ’’a;1 ’4d;1 ’4g;1
’1c;0 ’1b;0 ’’a;0 ’’a;0 ’4d;0 ’4g;0
Vc Vb Va Va Vd Vg
’’a;0 = 1 ’1a;0+ 4 ’4a;0 ’’a;1 = 1 ’1a;1+ 4 ’4a;1
1 + 4 1 + 4
1 min{ ’’a;0, ’’a;1} + 4 min{ ’’a;0, ’’a;1} + K
1 ’’1 + 4 ’’4 + rest

98. TRW Message Passing
Kolmogorov, 2006
’1c;1 ’1b;1 ’’a;1 ’’a;1 ’4d;1 ’4g;1
’1c;0 ’1b;0 ’’a;0 ’’a;0 ’4d;0 ’4g;0
Vc Vb Va Va Vd Vg
’’a;0 = 1 ’1a;0+ 4 ’4a;0 ’’a;1 = 1 ’1a;1+ 4 ’4a;1
1 + 4 1 + 4
( 1+ 4) min{ ’’a;0, ’’a;1} + K
1 ’’1 + 4 ’’4 + rest

99. TRW Message Passing
Kolmogorov, 2006
’1c;1 ’1b;1 ’’a;1 ’’a;1 ’4d;1 ’4g;1
’1c;0 ’1b;0 ’’a;0 ’’a;0 ’4d;0 ’4g;0
Vc Vb Va Va Vd Vg
min {p1+p2, q1+q2} ≥ min {p1, q1} + min {p2, q2}
( 1+ 4) min{ ’’a;0, ’’a;1} + K
1 ’’1 + 4 ’’4 + rest

100. TRW Message Passing
Kolmogorov, 2006
’1c;1 ’1b;1 ’’a;1 ’’a;1 ’4d;1 ’4g;1
’1c;0 ’1b;0 ’’a;0 ’’a;0 ’4d;0 ’4g;0
Vc Vb Va Va Vd Vg
Objective function increases or remains constant
( 1+ 4) min{ ’’a;0, ’’a;1} + K
1 ’’1 + 4 ’’4 + rest

101. TRW Message Passing
Initialize i. Take care of reparam constraint
Choose random variable Va
Compute min-marginals of Va for all trees
Node-average the min-marginals
REPEAT Can also do edge-averaging
Kolmogorov, 2006

102. Example 1
1 =1 2 =1 3 =1
l1
2 0 4 4 0 6 6 1 6
1 1 2 3 4 1
l0
5 0 2 2 1 3 3 0 4
Va Vb Vb Vc Vc Va
5 6 7
Pick variable Va. Reparameterize.

103. Example 1
1 =1 2 =1 3 =1
l1
5 -3 4 4 0 6 6 -3 10
-2 -1 2 3 1 -3
l0
7 -2 2 2 1 3 3 -3 7
Va Vb Vb Vc Vc Va
5 6 7
Average the min-marginals of Va

104. Example 1
1 =1 2 =1 3 =1
l1
7.5 -3 4 4 0 6 6 -3 7.5
-2 -1 2 3 1 -3
l0
7 -2 2 2 1 3 3 -3 7
Va Vb Vb Vc Vc Va
7 6 7
Pick variable Vb. Reparameterize.

105. Example 1
1 =1 2 =1 3 =1
l1
7.5 -7.5 8.5 9 -5 6 6 -3 7.5
-7 -5.5 -3 -1 1 -3
l0
7 -7 7 6 -3 3 3 -3 7
Va Vb Vb Vc Vc Va
7 6 7
Average the min-marginals of Vb

106. Example 1
1 =1 2 =1 3 =1
l1
7.5 -7.5 8.75 8.75 -5 6 6 -3 7.5
-7 -5.5 -3 -1 1 -3
l0
7 -7 6.5 6.5 -3 3 3 -3 7
Va Vb Vb Vc Vc Va
6.5 6.5 7
Value of dual does not increase

107. Example 1
1 =1 2 =1 3 =1
l1
7.5 -7.5 8.75 8.75 -5 6 6 -3 7.5
-7 -5.5 -3 -1 1 -3
l0
7 -7 6.5 6.5 -3 3 3 -3 7
Va Vb Vb Vc Vc Va
6.5 6.5 7
Maybe it will increase for Vc
NO

108. Example 1
1 =1 2 =1 3 =1
l1
7.5 -7.5 8.75 8.75 -5 6 6 -3 7.5
-7 -5.5 -3 -1 1 -3
l0
7 -7 6.5 6.5 -3 3 3 -3 7
Va Vb Vb Vc Vc Va
f1(a) = 0 f1(b) = 0 f2(b) = 0 f2(c) = 0 f3(c) = 0 f3(a) = 0
Strong Tree Agreement
Exact MAP Estimate

109. Example 2
1 =1 2 =1 3 =1
l1
2 0 2 0 1 0 0 0 4
1 1 0 0 1 1
l0
5 0 2 0 1 0 3 0 8
Va Vb Vb Vc Vc Va
4 0 4
Pick variable Va. Reparameterize.

110. Example 2
1 =1 2 =1 3 =1
l1
4 -2 2 0 1 0 0 0 4
-1 -1 0 0 0 1
l0
7 -2 2 0 1 0 3 -1 9
Va Vb Vb Vc Vc Va
4 0 4
Average the min-marginals of Va

111. Example 2
1 =1 2 =1 3 =1
l1
4 -2 2 0 1 0 0 0 4
-1 -1 0 0 0 1
l0
8 -2 2 0 1 0 3 -1 8
Va Vb Vb Vc Vc Va
4 0 4
Value of dual does not increase

112. Example 2
1 =1 2 =1 3 =1
l1
4 -2 2 0 1 0 0 0 4
-1 -1 0 0 0 1
l0
8 -2 2 0 1 0 3 -1 8
Va Vb Vb Vc Vc Va
4 0 4
Maybe it will decrease for Vb or Vc
NO

113. Example 2
1 =1 2 =1 3 =1
l1
4 -2 2 0 1 0 0 0 4
-1 -1 0 0 0 1
l0
8 -2 2 0 1 0 3 -1 8
Va Vb Vb Vc Vc Va
f1(a) = 1 f1(b) = 1 f2(b) = 1 f2(c) = 0 f3(c) = 1 f3(a) = 1
f2(b) = 0 f2(c) = 1
Weak Tree Agreement
Not Exact MAP Estimate

114. Example 2
1 =1 2 =1 3 =1
l1
4 -2 2 0 1 0 0 0 4
-1 -1 0 0 0 1
l0
8 -2 2 0 1 0 3 -1 8
Va Vb Vb Vc Vc Va
f1(a) = 1 f1(b) = 1 f2(b) = 1 f2(c) = 0 f3(c) = 1 f3(a) = 1
f2(b) = 0 f2(c) = 1
Weak Tree Agreement
Convergence point of TRW

115. Obtaining the Labeling
Only solves the dual. Primal solutions?
Va Vb Vc Fix the label
Of Va
Vd Ve Vf
Vg Vh Vi
’= i i

116. Obtaining the Labeling
Only solves the dual. Primal solutions?
Va Vb Vc Fix the label
Of Vb
Vd Ve Vf
Vg Vh Vi
’= i i
Continue in some fixed order
Meltzer et al., 2006

117. Outline
• Problem Formulation
• Reparameterization
• Belief Propagation
• Tree-reweighted Message Passing
– Integer Programming Formulation
– Linear Programming Relaxation and its Dual
– Convergent Solution for Dual
– Computational Issues and Theoretical Properties

118. Computational Issues of TRW
Basic Component is Belief Propagation
• Speed-ups for some pairwise potentials
Felzenszwalb & Huttenlocher, 2004
• Memory requirements cut down by half
Kolmogorov, 2006
• Further speed-ups using monotonic chains
Kolmogorov, 2006

119. Theoretical Properties of TRW
• Always converges, unlike BP
Kolmogorov, 2006
• Strong tree agreement implies exact MAP
Wainwright et al., 2001
• Optimal MAP for two-label submodular problems
ab;00 + ab;11 ≤ ab;01 + ab;10
Kolmogorov and Wainwright, 2005

120. Summary
• Trees can be solved exactly - BP
• No guarantee of convergence otherwise - BP
• Strong Tree Agreement - TRW-S
• Submodular energies solved exactly - TRW-S
• TRW-S solves an LP relaxation of MAP estimation