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Discrete Optimization Lecture 8

Discrete Optimization Lecture 8. M. Pawan Kumar pawan.kumar@ecp.fr. Slides available online http:// cvn.ecp.fr /personnel/ pawan /. Recap !!. Questions?. Energy Minimization. 0. 6. 1. 3. 2. 0. 4. Label l 1. 1. 2. 3. 4. 1. 1. Label l 0. 1. 0. 5. 0. 3. 7. 2. V b. V c.

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Discrete Optimization Lecture 8

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  1. Discrete OptimizationLecture 8 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online http://cvn.ecp.fr/personnel/pawan/

  2. Recap !! Questions?

  3. Energy Minimization 0 6 1 3 2 0 4 Label l1 1 2 3 4 1 1 Label l0 1 0 5 0 3 7 2 Vb Vc Vd Va q* = min Q(f; ) = Q(f*; ) Q(f; ) = ∑a a;f(a) + ∑(a,b) ab;f(a)f(b) f* = arg min Q(f; )

  4. Min-Marginals 0 6 1 3 2 0 4 Label l1 1 2 3 4 1 1 Label l0 1 0 5 0 3 7 2 Vb Vc Vd Va such that f(a) = i f* = arg min Q(f; ) Min-marginal qa;i

  5. Min-Marginals and MAP • Minimum min-marginal of any variable = • energy of MAP labelling

  6. Equivalently + Mba;i ’a;i = a;i + Mab;k ’ab;ik = ab;ik - Mab;k - Mba;i Reparameterization ’ is a reparameterization of , iff ’   Q(f; ’) = Q(f; ), for all f Kolmogorov, PAMI, 2006 ’b;k = b;k

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

  8. Outline • LP Relaxation and its Dual • TRW Message Passing • Dual Decomposition

  9. 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

  10. Dual of the LP Relaxation Wainwright et al., 2001 1 Va Vb Vc Va Vb Vc 2 Vd Ve Vf Vd Ve Vf 3 Vg Vh Vi Vg Vh Vi 4 5 6  Va Vb Vc Vd Ve Vf Vg Vh Vi  i  

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

  12. Outline • LP Relaxation and its Dual • TRW Message Passing • Dual Decomposition

  13. Things to Remember • BP is exact for trees • Every iteration provides a reparameterization • Forward-pass computes min-marginals of root

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

  15. 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  q*(i)  i  

  16. 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 Reparameterize to obtain min-marginals of Va q*(1) + q*(4) + K 1 +4 + rest 

  17. 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 q*(’1) + q*(’4) + K ’1 +’4 + rest

  18. 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 q*(’1) + q*(’4) + K ’1 +’4 + rest 

  19. 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 min{’1a;0,’1a;1} + min{’4a;0,’4a;1} + K ’1 +’4 + rest 

  20. 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 average of min-marginals of Va min{’1a;0,’1a;1} + min{’4a;0,’4a;1} + K ’1 +’4 + rest 

  21. 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 = ’1a;0+ ’4a;0 ’’a;1 = ’1a;1+ ’4a;1 2 2 min{’1a;0,’1a;1} + min{’4a;0,’4a;1} + K ’1 +’4 + rest 

  22. 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 = ’1a;0+ ’4a;0 ’’a;1 = ’1a;1+ ’4a;1 2 2 min{’1a;0,’1a;1} + min{’4a;0,’4a;1} + K ’’1 +’’4 + rest

  23. 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 = ’1a;0+ ’4a;0 ’’a;1 = ’1a;1+ ’4a;1 2 2 min{’1a;0,’1a;1} + min{’4a;0,’4a;1} + K ’’1 +’’4 + rest  

  24. 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 = ’1a;0+ ’4a;0 ’’a;1 = ’1a;1+ ’4a;1 2 2 2 min{’’a;0, ’’a;1} + K ’’1 +’’4 + rest  

  25. 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} 2 min{’’a;0, ’’a;1} + K ’’1 +’’4 + rest  

  26. 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 2 min{’’a;0, ’’a;1} + K ’’1 +’’4 + rest  

  27. 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 Can also do edge-averaging REPEAT Kolmogorov, 2006

  28. Outline • LP Relaxation and its Dual • TRW Message Passing • Examples • Primal Solution • Results • Dual Decomposition

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

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

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

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

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

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

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

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

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

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

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

  40. Example 2 4 -2 2 0 1 0 0 0 4 l1 0 -1 0 1 -1 0 l0 8 -2 0 1 3 -1 2 0 8 Vb Vc Va Va Vb Vc 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

  41. Example 2 4 -2 2 0 1 0 0 0 4 l1 0 -1 0 1 -1 0 l0 8 -2 0 1 3 -1 2 0 8 Vb Vc Va Va Vb Vc 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

  42. Outline • LP Relaxation and its Dual • TRW Message Passing • Examples • Primal Solution • Results • Dual Decomposition

  43. Obtaining the Labelling Only solves the dual. Primal solutions? Fix the label Of Va Va Vb Vc Vd Ve Vf Vg Vh Vi ’=  i  

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

  45. 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

  46. 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

  47. Outline • LP Relaxation and its Dual • TRW Message Passing • Examples • Primal Solutions • Results • Dual Decomposition

  48. Results Szeliski et al. , 2008 Binary Segmentation Labels - {foreground, background} Unary Potentials: -log(likelihood) using learnt fg/bg models Pairwise Potentials: 0, if same labels 1 - exp(|da- db|), if different labels

  49. Results Szeliski et al. , 2008 Binary Segmentation TRW Labels - {foreground, background} Unary Potentials: -log(likelihood) using learnt fg/bg models Pairwise Potentials: 0, if same labels 1 - exp(|da- db|), if different labels

  50. Results Szeliski et al. , 2008 Binary Segmentation Belief Propagation Labels - {foreground, background} Unary Potentials: -log(likelihood) using learnt fg/bg models Pairwise Potentials: 0, if same labels 1 - exp(|da- db|), if different labels

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