Discrete Inference and Learning Lecture 3

Transcription

1 Discrete Inference and Learning Lecture 3 MVA h<p://thoth.inrialpes.fr/~alahari/disinflearn Slides based on material from Pushmeet Kohli, Nikos Komodakis, M. Pawan Kumar

2 Recap The st-mincut problem Connection between st-mincut and energy minimization? What problems can we solve using st-mincut? st-mincut based Move algorithms

3 St-mincut based Move algorithms E(y) = θ i (y i ) + θ ij (y i,y j ) i i,j y ϵ Labels L = {l 1, l 2,, l k } Commonly used for solving non-submodular multi-label problems Extremely efficient and produce good solutions Not Exact: Produce local optima

4 Move Making Algorithms Current Solution Search Neighbourhood Optimal Move Energy Solution Space

5 Move Making Algorithms Current Solution Search Neighbourhood Optimal Move Energy Solution Space

6 Computing the Optimal Move Energy (t) y c Current Solution Search Neighbourhood Optimal Move Key Property Move Space Solution Space Bigger move space Better solutions Finding the optimal move hard

7 Moves using Graph Cuts Expansion and Swap move algorithms [Boykov Veksler and Zabih, PAMI 2001] Makes a series of changes to the solution (moves) Each move results in a solution with smaller energy Move Space (t) : 2 N Space of Solutions (y) : L N N L Current Solution Search Neighbourhood Number of Variables Number of Labels

8 Moves using Graph Cuts Expansion and Swap move algorithms [Boykov Veksler and Zabih, PAMI 2001] Makes a series of changes to the solution (moves) Each move results in a solution with smaller energy Current Solution Move to new solution Construct a move function Minimize move function to get optimal move How to minimize move functions?

9 General Binary Moves y = t y 1 + (1- t) y 2 New solution Current Solution Second solution E m (t) = E(t y 1 + (1- t) y 2 ) Minimize over move variables t to get the optimal move Move energy is a submodular QPBF (Exact Minimization Possible) Boykov, Veksler and Zabih, PAMI 2001

10 Expansion Move Variables take label α or retain current label Status: Initialize with Tree Tree Ground House Sky [Boykov, Veksler, Zabih]

11 Expansion Move Variables take label α or retain current label Status: Expand Ground Tree Ground House Sky [Boykov, Veksler, Zabih]

12 Expansion Move Variables take label α or retain current label Status: Expand House Tree Ground House Sky [Boykov, Veksler, Zabih]

13 Expansion Move Variables take label α or retain current label Status: Expand Sky Tree Ground House Sky [Boykov, Veksler, Zabih]

14 Expansion Move Variables take label α or retain current label Move energy is submodular if: Unary Potentials: Arbitrary Pairwise potentials: Metric θ ij (l a,l b ) 0 θ ij (l a,l b ) = 0 iff a = b Semi metric Examples: Potts model, Truncated linear Cannot solve truncated quadratic [Boykov, Veksler, Zabih]

15 Expansion Move Variables take label α or retain current label Move energy is submodular if: Unary Potentials: Arbitrary Pairwise potentials: Metric θ ij (l a,l b ) + θ ij (l b,l c ) θ ij (l a,l c ) Triangle Inequality Examples: Potts model, Truncated linear Cannot solve truncated quadratic [Boykov, Veksler, Zabih]

16 Swap Move Variables labeled α, β can swap their labels Swap Sky, House Tree Ground House Sky [Boykov, Veksler, Zabih]

17 Swap Move Variables labeled α, β can swap their labels Swap Sky, House Tree Ground House Sky [Boykov, Veksler, Zabih]

18 Swap Move Variables labeled α, β can swap their labels Move energy is submodular if: Unary Potentials: Arbitrary Pairwise potentials: Semimetric θ ij (l a,l b ) 0 θ ij (l a,l b ) = 0 a = b Examples: Potts model, Truncated Convex [Boykov, Veksler, Zabih]

19 Summary Labelling Problem Exact Transformation (global optimum) Or Relaxed transformation (partially optimal) Submodular Quadratic Pseudoboolean Function S st-mincut Sub-problem T Move making algorithms

20 Where do we stand? Grid graph - submodular, 2-label: Use graphcuts metric : Use expansion otherwise: Use TRW, dual decomposition, relaxation Chain/Tree, 2/multi-label: Use BP

21 Dynamic Energy Minimization

22 Image Segmentation in Video n-links s = 0 st-cut E(x) x * w pq t = 1 Video frame Flow Global Optimum

23 Image Segmentation in Video Video frame Flow Global Optimum

24 Dynamic Energy Minimization E A minimize S A Can we do better? Recycling Solutions E B computationally expensive operation S B Boykov & Jolly ICCV 01, Kohli & Torr (ICCV05, PAMI07)

25 Dynamic Energy Minimization E A minimize S A Reuse flow A and B similar differences between A and B E B* Simpler energy cheaper operation E B computationally expensive operation S B Reparametrization Boykov & Jolly ICCV 01, Kohli Kohli & Torr & Torr (ICCV05, (ICCV05, PAMI07) PAMI07)

26 Dynamic Energy Minimization Original Energy E(a 1,a 2 ) = 2a 1 + 5ā 1 + 9a 2 + 4ā 2 + 2a 1 ā 2 + ā 1 a 2 Reparameterized Energy E(a 1,a 2 ) = 8 + ā 1 + 3a 2 + 3ā 1 a 2 New Energy E(a 1,a 2 ) = 2a 1 + 5ā 1 + 9a 2 + 4ā 2 + 7a 1 ā 2 + ā 1 a 2 New Reparameterized Energy E(a 1,a 2 ) = 8 + ā 1 + 3a 2 + 3ā 1 a 2 + 5a 1 ā 2 Kohli & Torr (ICCV05, PAMI07) Boykov & Jolly ICCV 01, Kohli & Torr (ICCV05, PAMI07)

27 Outline Reparameterization (lecture 1) Belief Propagation (lecture 1) Tree-reweighted Message Passing Integer Programming Formulation Linear Programming Relaxation and its Dual Convergent Solution for Dual Computational Issues and Theoretical Properties

28 First Recap of Integer Linear Program

29 Integer Linear Program max x c T x s.t. A x b x is an integer vector Every element of x is an integer

30 Integer Linear Program max x c T x s.t. A x b x Z n Every element of x is an integer

31 Example max x x 1 + x 2 s.t. x 1 0 x 2 0 4x 1 x 2 8 2x 1 + x x 1-2x 2-2 x Z n

32 Example x 1 0 x 2 0

33 Example x 1 0 x 2 0 4x 1 x 2 = 8

34 Example x 1 0 x 2 0 4x 1 x 2 8

35 Example x 1 0 x 2 0 4x 1 x 2 8 2x 1 + x x 1-2x 2-2

36 Example (2,6) (3,4) x 1 0 x 2 0 4x 1 x 2 8 2x 1 + x 2 10 (0,1) (0,0) (2,0) 5x 1-2x 2-2 x Z n max x c T x

37 Example (2,6) (3,4) x 1 0 x 2 0 4x 1 x 2 8 2x 1 + x 2 10 (0,1) (0,0) Why? (2,0) True in general? 5x 1-2x 2-2 x Z n max x c T x

38 Why is the soluaon at a vertex?

39 Integer Programming Formulation Unary Potentials Label l θ a;0 = 5 θ b;0 = θ a;1 = 2 θ b;1 = 4 Label l V a V b Labeling f(a) = 1 f(b) = 0 y a;0 = 0 y a;1 = 1 y b;0 = 1 y b;1 = 0 Any f(.) has equivalent boolean variables y a;i

40 Integer Programming Formulation Unary Potentials Label l θ a;0 = 5 θ b;0 = θ a;1 = 2 θ b;1 = 4 Label l V a V b Labeling f(a) = 1 f(b) = 0 y a;0 = 0 y a;1 = 1 y b;0 = 1 y b;1 = 0 Find the optimal variables y a;i

41 Integer Programming Formulation Unary Potentials Label l θ a;0 = 5 θ b;0 = θ a;1 = 2 θ b;1 = 4 Label l V a V b Sum of Unary Potentials a i θ a;i y a;i y a;i {0,1}, for all V a, l i i y a;i = 1, for all V a

42 Integer Programming Formulation Pairwise Potentials Label l θ ab;00 = 0 θ ab;01 = θ ab;10 = 1 θ ab;11 = 0 Label l V a V b Sum of Pairwise Potentials (a,b) ik θ ab;ik y a;i y b;k y a;i {0,1} i y a;i = 1

43 Integer Programming Formulation Pairwise Potentials Label l θ ab;00 = 0 θ ab;01 = θ ab;10 = 1 θ ab;11 = 0 Label l V a V b Sum of Pairwise Potentials (a,b) ik θ ab;ik y ab;ik y a;i {0,1} y ab;ik = y a;i y b;k i y a;i = 1

44 Integer Programming Formulation min a i θ a;i y a;i + (a,b) ik θ ab;ik y ab;ik y a;i {0,1} i y a;i = 1 y ab;ik = y a;i y b;k

45 Integer Programming Formulation min θ T y y a;i {0,1} i y a;i = 1 y ab;ik = y a;i y b;k θ = [ θ a;i. ; θ ab;ik.] y = [ y a;i. ; y ab;ik.]

46 Integer Programming Formulation min θ T y y a;i {0,1} i y a;i = 1 y ab;ik = y a;i y b;k Solve to obtain MAP labeling y*

47 Integer Programming Formulation min θ T y y a;i {0,1} i y a;i = 1 y ab;ik = y a;i y b;k But we can t solve it in general

48 Outline Reparameterization (lecture 1) Belief Propagation (lecture 1) Tree-reweighted Message Passing Integer Programming Formulation Linear Programming Relaxation and its Dual Convergent Solution for Dual Computational Issues and Theoretical Properties

49 Linear Programming Relaxation min θ T y y a;i {0,1} i y a;i = 1 y ab;ik = y a;i y b;k Two reasons why we can t solve this

50 Linear Programming Relaxation min θ T y y a;i [0,1] i y a;i = 1 y ab;ik = y a;i y b;k One reason why we can t solve this

51 Linear Programming Relaxation min θ T y y a;i [0,1] i y a;i = 1 k y ab;ik = k y a;i y b;k One reason why we can t solve this

52 Linear Programming Relaxation min θ T y y a;i [0,1] i y a;i = 1 k y ab;ik = y a;i k y b;k = 1 One reason why we can t solve this

53 Linear Programming Relaxation min θ T y y a;i [0,1] i y a;i = 1 k y ab;ik = y a;i One reason why we can t solve this

54 Linear Programming Relaxation min θ T y y a;i [0,1] i y a;i = 1 k y ab;ik = y a;i No reason why we can t solve this * *memory requirements, time complexity

55 Wainwright et al., 2001 Dual of the LP Relaxation V a V b V c min θ T y V d V e V f y a;i [0,1] V g V h V i θ i y a;i = 1 k y ab;ik = y a;i

56 Wainwright et al., 2001 V a V b V c V d V e V f V g V h V i Dual of the LP Relaxation θ 1 θ 2 θ 3 θ ρ i 0 V a V b V c V d V e V f V g V h V i ρ 1 ρ 2 ρ 3 θ 4 θ 5 θ 6 V a V b V c V d V e V f ρ i θ i = θ V g V h V i ρ 4 ρ 5 ρ 6

57 Wainwright et al., 2001 V a V b V c V d V e V f V g V h V i θ Dual of LP max ρ i q*(θ i ) ρ i θ i = θ Dual of the LP Relaxation q*(θ 1 ) q*(θ 2 ) q*(θ 3 ) ρ i 0 V a V b V c V d V e V f V g V h V i ρ 1 ρ 2 ρ 3 q*(θ 4 ) q*(θ 5 ) q*(θ 6 ) V a V b V c V d V e V f V g V h V i ρ 4 ρ 5 ρ 6

58 Wainwright et al., 2001 V a V b V c V d V e V f V g V h V i θ Dual of LP max ρ i q*(θ i ) ρ i θ i θ Dual of the LP Relaxation q*(θ 1 ) q*(θ 2 ) q*(θ 3 ) ρ i 0 V a V b V c V d V e V f V g V h V i ρ 1 ρ 2 ρ 3 q*(θ 4 ) q*(θ 5 ) q*(θ 6 ) V a V b V c V d V e V f V g V h V i ρ 4 ρ 5 ρ 6

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

60 Outline Reparameterization (lecture 1) Belief Propagation (lecture 1) Tree-reweighted Message Passing Integer Programming Formulation Linear Programming Relaxation and its Dual Convergent Solution for Dual Computational Issues and Theoretical Properties

61 Kolmogorov, 2006 θ 1 V a V b V c TRW Message Passing ρ 1 θ 4 θ 5 θ 6 V a V b V c θ 2 V d V e V f ρ 2 V d V e V f θ 3 V g V h V i ρ 3 V g V h V i ρ 4 ρ 5 ρ 6 Pick a variable V a ρ i q*(θ i ) ρ i θ i θ

62 Kolmogorov, 2006 TRW Message Passing θ 1 c;1 θ 1 b;1 θ 1 a;1 θ 4 a;1 θ 4 d;1 θ 4 g;1 θ 1 c;0 θ 1 b;0 θ 1 a;0 θ 4 a;0 θ 4 d;0 θ 4 g;0 V c V b V a V a V d V g ρ i q*(θ i ) ρ i θ i θ

63 Kolmogorov, 2006 TRW Message Passing θ 1 c;1 θ 1 b;1 θ 1 a;1 θ 4 a;1 θ 4 d;1 θ 4 g;1 θ 1 c;0 θ 1 b;0 θ 1 a;0 θ 4 a;0 θ 4 d;0 θ 4 g;0 V c V b V a V a V d V g Reparameterize to obtain min-marginals of V a ρ 1 q*(θ 1 ) + ρ 4 q*(θ 4 ) + K ρ 1 θ 1 + ρ 4 θ 4 + θ rest θ

64 Kolmogorov, 2006 TRW Message Passing θ 1 c;1 θ 1 b;1 θ 1 a;1 θ 4 a;1 θ 4 d;1 θ 4 g;1 θ 1 c;0 θ 1 b;0 θ 1 a;0 θ 4 a;0 θ 4 d;0 θ 4 g;0 V c V b V a V a V d V g One pass of Belief Propagation ρ 1 q*(θ 1 ) + ρ 4 q*(θ 4 ) + K ρ 1 θ 1 + ρ 4 θ 4 + θ rest

65 Kolmogorov, 2006 TRW Message Passing θ 1 c;1 θ 1 b;1 θ 1 a;1 θ 4 a;1 θ 4 d;1 θ 4 g;1 θ 1 c;0 θ 1 b;0 θ 1 a;0 θ 4 a;0 θ 4 d;0 θ 4 g;0 V c V b V a V a V d V g Remain the same ρ 1 q*(θ 1 ) + ρ 4 q*(θ 4 ) + K ρ 1 θ 1 + ρ 4 θ 4 + θ rest θ

66 Kolmogorov, 2006 TRW Message Passing θ 1 c;1 θ 1 b;1 θ 1 a;1 θ 4 a;1 θ 4 d;1 θ 4 g;1 θ 1 c;0 θ 1 b;0 θ 1 a;0 θ 4 a;0 θ 4 d;0 θ 4 g;0 V c V b V a V a V d V g ρ 1 min{θ 1 a;0,θ 1 a;1 } + ρ4 min{θ 4 a;0,θ 4 a;1 } + K ρ 1 θ 1 + ρ 4 θ 4 + θ rest θ

67 Kolmogorov, 2006 TRW Message Passing θ 1 c;1 θ 1 b;1 θ 1 a;1 θ 4 a;1 θ 4 d;1 θ 4 g;1 θ 1 c;0 θ 1 b;0 θ 1 a;0 θ 4 a;0 θ 4 d;0 θ 4 g;0 V c V b V a V a V d V g Compute weighted average of min-marginals of V a ρ 1 min{θ 1 a;0,θ 1 a;1 } + ρ4 min{θ 4 a;0,θ 4 a;1 } + K ρ 1 θ 1 + ρ 4 θ 4 + θ rest θ

68 Kolmogorov, 2006 TRW Message Passing θ 1 c;1 θ 1 b;1 θ 1 a;1 θ 4 a;1 θ 4 d;1 θ 4 g;1 θ 1 c;0 θ 1 b;0 θ 1 a;0 θ 4 a;0 θ 4 d;0 θ 4 g;0 V c V b V a V a V d V g θ a;0 = ρ 1 θ 1 a;0 + ρ4 θ 4 a;0 θ a;1 = ρ 1 θ 1 a;1 + ρ4 θ 4 a;1 ρ 1 + ρ 4 ρ 1 + ρ 4 ρ 1 min{θ 1 a;0,θ 1 a;1 } + ρ4 min{θ 4 a;0,θ 4 a;1 } + K ρ 1 θ 1 + ρ 4 θ 4 + θ rest θ

69 Kolmogorov, 2006 TRW Message Passing θ 1 c;1 θ 1 b;1 θ a;1 θ a;1 θ 4 d;1 θ 4 g;1 θ 1 c;0 θ 1 b;0 θ a;0 θ a;0 θ 4 d;0 θ 4 g;0 V c V b V a V a V d V g θ a;0 = ρ 1 θ 1 a;0 + ρ4 θ 4 a;0 θ a;1 = ρ 1 θ 1 a;1 + ρ4 θ 4 a;1 ρ 1 + ρ 4 ρ 1 + ρ 4 ρ 1 min{θ 1 a;0,θ 1 a;1 } + ρ4 min{θ 4 a;0,θ 4 a;1 } + K ρ 1 θ 1 + ρ 4 θ 4 + θ rest

70 Kolmogorov, 2006 TRW Message Passing θ 1 c;1 θ 1 b;1 θ a;1 θ a;1 θ 4 d;1 θ 4 g;1 θ 1 c;0 θ 1 b;0 θ a;0 θ a;0 θ 4 d;0 θ 4 g;0 V c V b V a V a V d V g θ a;0 = ρ 1 θ 1 a;0 + ρ4 θ 4 a;0 θ a;1 = ρ 1 θ 1 a;1 + ρ4 θ 4 a;1 ρ 1 + ρ 4 ρ 1 + ρ 4 ρ 1 min{θ 1 a;0,θ 1 a;1 } + ρ4 min{θ 4 a;0,θ 4 a;1 } + K ρ 1 θ 1 + ρ 4 θ 4 + θ rest θ

71 Kolmogorov, 2006 TRW Message Passing θ 1 c;1 θ 1 b;1 θ a;1 θ a;1 θ 4 d;1 θ 4 g;1 θ 1 c;0 θ 1 b;0 θ a;0 θ a;0 θ 4 d;0 θ 4 g;0 V c V b V a V a V d V g θ a;0 = ρ 1 θ 1 a;0 + ρ4 θ 4 a;0 θ a;1 = ρ 1 θ 1 a;1 + ρ4 θ 4 a;1 ρ 1 + ρ 4 ρ 1 + ρ 4 ρ 1 min{θ a;0,θ a;1 } + ρ 4 min{θ a;0,θ a;1 } + K ρ 1 θ 1 + ρ 4 θ 4 + θ rest θ

72 Kolmogorov, 2006 TRW Message Passing θ 1 c;1 θ 1 b;1 θ a;1 θ a;1 θ 4 d;1 θ 4 g;1 θ 1 c;0 θ 1 b;0 θ a;0 θ a;0 θ 4 d;0 θ 4 g;0 V c V b V a V a V d V g θ a;0 = ρ 1 θ 1 a;0 + ρ4 θ 4 a;0 θ a;1 = ρ 1 θ 1 a;1 + ρ4 θ 4 a;1 ρ 1 + ρ 4 ρ 1 + ρ 4 (ρ 1 + ρ 4 ) min{θ a;0, θ a;1 } + K ρ 1 θ 1 + ρ 4 θ 4 + θ rest θ

73 Kolmogorov, 2006 TRW Message Passing θ 1 c;1 θ 1 b;1 θ a;1 θ a;1 θ 4 d;1 θ 4 g;1 θ 1 c;0 θ 1 b;0 θ a;0 θ a;0 θ 4 d;0 θ 4 g;0 V c V b V a V a V d V g min {p 1 +p 2, q 1 +q 2 } min {p 1, q 1 } + min {p 2, q 2 } (ρ 1 + ρ 4 ) min{θ a;0, θ a;1 } + K ρ 1 θ 1 + ρ 4 θ 4 + θ rest θ

74 Kolmogorov, 2006 TRW Message Passing θ 1 c;1 θ 1 b;1 θ a;1 θ a;1 θ 4 d;1 θ 4 g;1 θ 1 c;0 θ 1 b;0 θ a;0 θ a;0 θ 4 d;0 θ 4 g;0 V c V b V a V a V d V g Objective function increases or remains constant (ρ 1 + ρ 4 ) min{θ a;0, θ a;1 } + K ρ 1 θ 1 + ρ 4 θ 4 + θ rest θ

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

76 Example 1 l 1 2 ρ 1 = ρ 2 =1 ρ 3 = l V a V b V b V c V c V a Pick variable V a. Reparameterize.

77 Example 1 l 1 5 ρ 1 = ρ 2 =1 ρ 3 = l V a V b V b V c V c V a Average the min-marginals of V a

78 Example 1 l ρ 1 = ρ 2 =1 ρ 3 = l V a V b V b V c V c V a Pick variable V b. Reparameterize.

79 Example 1 l ρ 1 = ρ 2 =1 ρ 3 = l V a V b V b V c V c V a Average the min-marginals of V b

80 Example 1 l ρ 1 = ρ 2 =1 ρ 3 = l V a V b V b V c V c V a Value of dual does not increase

81 Example 1 l ρ 1 = ρ 2 =1 ρ 3 = l V a V b V b V c V c V a Maybe it will increase for V c NO

82 Example 1 l ρ 1 = ρ 2 =1 ρ 3 = l V a V b V b V c V c V a f 1 (a) = 0 f 1 (b) = 0 f 2 (b) = 0 f 2 (c) = 0 f 3 (c) = 0 f 3 (a) = 0 Strong Tree Agreement Exact MAP Estimate

83 Example 2 l 1 2 ρ 1 = ρ 2 =1 ρ 3 = l V a V b V b V c V c V a Pick variable V a. Reparameterize.

84 Example 2 l 1 4 ρ 1 = ρ 2 =1 ρ 3 = l V a V b V b V c V c V a Average the min-marginals of V a

85 Example 2 l 1 4 ρ 1 = ρ 2 =1 ρ 3 = l V a V b V b V c V c V a Value of dual does not increase

86 Example 2 l 1 4 ρ 1 = ρ 2 =1 ρ 3 = l V a V b V b V c V c V a Maybe it will decrease for V b or V c NO

87 Example 2 l 1 4 ρ 1 = ρ 2 =1 ρ 3 = l V a V b V b V c V c V a f 1 (a) = 1 f 1 (b) = 1 f 2 (b) = 1 f 2 (c) = 0 f 3 (c) = 1 f 3 (a) = 1 f 2 (b) = 0 f 2 (c) = 1 Weak Tree Agreement Not Exact MAP Estimate

88 Example 2 l 1 4 ρ 1 = ρ 2 =1 ρ 3 = l V a V b V b V c V c V a f 1 (a) = 1 f 1 (b) = 1 f 2 (b) = 1 f 2 (c) = 0 f 3 (c) = 1 f 3 (a) = 1 f 2 (b) = 0 f 2 (c) = 1 Weak Tree Agreement Convergence point of TRW

89 Obtaining the Labeling Only solves the dual. Primal solutions? V a V b V c V d V e V f Fix the label of V a V g V h V i θ = ρ i θ i θ

90 Obtaining the Labeling Only solves the dual. Primal solutions? V a V b V c V d V e V f Fix the label of V b V g V h V i θ = ρ i θ i θ Continue in some fixed order Meltzer et al., 2006

91 Outline Reparameterization (lecture 1) Belief Propagation (lecture 1) Tree-reweighted Message Passing Integer Programming Formulation Linear Programming Relaxation and its Dual Convergent Solution for Dual Computational Issues and Theoretical Properties

92 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

93 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 θ ab;00 + θ ab;11 θ ab;01 + θ ab;10 Kolmogorov and Wainwright, 2005

94 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