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Relaxations and Moves for MAP Estimation in MRFs

S T A N F O R D. Relaxations and Moves for MAP Estimation in MRFs. M. Pawan Kumar. Vladimir Kolmogorov. Philip Torr. Daphne Koller. Our Problem. 0. 6. 1. 3. 2. 0. 4. Label l 2. 1. 2. 4. 1. 1. 3. Label l 1. 1. 0. 5. 0. 3. 7. 2. v 1. v 2. v 3. v 4.

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Relaxations and Moves for MAP Estimation in MRFs

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  1. S T A N F O R D Relaxations and Moves forMAP Estimation in MRFs M. Pawan Kumar Vladimir Kolmogorov Philip Torr Daphne Koller

  2. Our Problem 0 6 1 3 2 0 4 Label l2 1 2 4 1 1 3 Label l1 1 0 5 0 3 7 2 v1 v2 v3 v4 Random Variables V = {v1, ... ,v4} Label Set L = {l1, l2} Labeling f: V  L (shown in red)

  3. Our Problem 0 6 1 3 2 0 4 Label l2 1 2 4 1 1 3 Label l1 1 0 5 0 3 7 2 v1 v2 v3 v4 Random Variables V = {v1, ... ,v4} Label Set L = {l1, l2} Labeling f: V  L (shown in red) Energy of Labeling E(f) = 13 (shown in green)

  4. Our Problem 0 6 1 3 2 0 4 Label l2 1 2 4 1 1 3 Label l1 1 0 5 0 3 7 2 v1 v2 v3 v4 Find f* = argminf E(f) Arbitrary topology, discrete label set, potentials (NP-hard) Pairwise energy function: unary and pairwise potentials (still NP-hard)

  5. Outline • Convex Relaxations • Integer Programming Formulation • LP Relaxation • SDP Relaxation • SOCP Relaxation • Comparing Relaxations • Move Making Algorithms • Some Interesting Open Problems

  6. Cost of v1 = 2 Cost of v1 = 1 Integer Programming Formulation 2 0 4 Unary Potentials Label l2 1 3 Label l1 5 0 2 v1 v2 Labeling f shown in red ; 2 4 ] 2 Unary Potential u = [ 5

  7. v1= 2 v1 1 Integer Programming Formulation 2 0 4 Unary Potentials Label l2 1 3 Label l1 5 0 2 v1 v2 Labeling f shown in red ; 2 4 ] 2 Unary Potential u = [ 5 Label vector x = [ -1 1 ; 1 -1 ]T Recall that the aim is to find the optimal x

  8. Integer Programming Formulation 2 0 4 Unary Potentials Label l2 1 3 Label l1 5 0 2 v1 v2 Labeling f shown in red ; 2 4 ] 2 Unary Potential u = [ 5 Label vector x = [ -1 1 ; 1 -1 ]T 1 Sum of Unary Potentials = ∑iui (1 + xi) 2

  9. Pairwise Potential P Cost of v1 = 1 and v1 = 1 0 Cost of v1 = 1 and v2 = 1 0 0 1 0 Cost of v1 = 1 and v2 = 2 0 1 0 0 3 0 0 0 Integer Programming Formulation 2 0 4 Pairwise Potentials Label l2 1 3 Label l1 5 0 2 v1 v2 Labeling f shown in red 0 3 0

  10. Pairwise Potential P 0 0 0 1 0 0 1 0 0 3 0 0 0 Integer Programming Formulation 2 0 4 Pairwise Potentials Label l2 1 3 Label l1 5 0 2 v1 v2 Labeling f shown in red Sum of Pairwise Potentials 1 ∑ijPij (1 + xi)(1+xj) 0 3 0 4

  11. Pairwise Potential P 0 0 0 1 0 1 = ∑ijPij (1 + xi + xj + Xij) 4 0 1 0 0 3 0 0 0 Integer Programming Formulation 2 0 4 Pairwise Potentials Label l2 1 3 Label l1 5 0 2 v1 v2 Labeling f shown in red Sum of Pairwise Potentials 1 ∑ijPij (1 + xi +xj + xixj) 0 3 0 4 X = x xT Xij = xi xj

  12. Uniqueness Constraint ∑ xi = 2 - |L| i  va Integer Programming Formulation Constraints • Integer Constraints xi{-1,1} X = x xT

  13. ∑ xi = 2 - |L| i  va Non-Convex Integer Programming Formulation 1 1 ∑ Pij (1 + xi + xj + Xij) x* = argmin + ∑ ui (1 + xi) 4 2 Convex xi{-1,1} X = x xT

  14. Outline • Convex Relaxations • Integer Programming Formulation • LP Relaxation • SDP Relaxation • SOCP Relaxation • Comparing Relaxations • Move Making Algorithms • Some Interesting Open Problems

  15. ∑ xi = 2 - |L| i  va LP Relaxation Schlesinger, 1976 Retain Convex Part 1 1 ∑ Pij (1 + xi + xj + Xij) x* = argmin + ∑ ui (1 + xi) 4 2 Relax Non-Convex Constraint xi{-1,1} X = x xT

  16. ∑ xi = 2 - |L| i  va LP Relaxation Schlesinger, 1976 Retain Convex Part 1 1 ∑ Pij (1 + xi + xj + Xij) x* = argmin + ∑ ui (1 + xi) 4 2 xi[-1,1] Relax Non-Convex Constraint X = x xT

  17. ∑ Xij = (2 - |L|) xi j  vb LP Relaxation Schlesinger, 1976 X = x xT Xij[-1,1] 1 + xi + xj + Xij≥ 0

  18. ∑ xi = 2 - |L| i  va LP Relaxation Schlesinger, 1976 Retain Convex Part 1 1 ∑ Pij (1 + xi + xj + Xij) x* = argmin + ∑ ui (1 + xi) 4 2 xi[-1,1] Relax Non-Convex Constraint X = x xT

  19. ∑ xi = 2 - |L| i  va ∑ Xij = (2 - |L|) xi j  vb LP Relaxation Schlesinger, 1976 Retain Convex Part 1 1 ∑ Pij (1 + xi + xj + Xij) x* = argmin + ∑ ui (1 + xi) 4 2 xi[-1,1], Xij[-1,1] 1 + xi + xj + Xij≥ 0

  20. Outline • Convex Relaxations • Integer Programming Formulation • LP Relaxation • SDP Relaxation • SOCP Relaxation • Comparing Relaxations • Move Making Algorithms • Some Interesting Open Problems

  21. ∑ xi = 2 - |L| i  va SDP Relaxation Lasserre, 2000 Retain Convex Part 1 1 ∑ Pij (1 + xi + xj + Xij) x* = argmin + ∑ ui (1 + xi) 4 2 Relax Non-Convex Constraint xi{-1,1} X = x xT

  22. ∑ xi = 2 - |L| i  va SDP Relaxation Lasserre, 2000 Retain Convex Part 1 1 ∑ Pij (1 + xi + xj + Xij) x* = argmin + ∑ ui (1 + xi) 4 2 xi[-1,1] Relax Non-Convex Constraint X = x xT

  23. 1 xT = x X Non-Convex SDP Relaxation . . . 1 1 x1 x2 xn x1 x2 . . . xn Xii = 1 Convex Positive Semidefinite Rank = 1

  24. 1 xT = x X Convex SDP Relaxation . . . 1 1 x1 x2 xn x1 x2 . . . xn Xii = 1 Positive Semidefinite

  25. ∑ xi = 2 - |L| i  va SDP Relaxation Lasserre, 2000 Retain Convex Part 1 1 ∑ Pij (1 + xi + xj + Xij) x* = argmin + ∑ ui (1 + xi) 4 2 xi[-1,1] Relax Non-Convex Constraint X = x xT

  26. ∑ xi = 2 - |L| i  va X - xxT 0 SDP Relaxation Lasserre, 2000 Retain Convex Part 1 1 ∑ Pij (1 + xi + xj + Xij) x* = argmin + ∑ ui (1 + xi) 4 2 xi[-1,1] Positive Semidefinite Xii = 1 Inefficient Accurate

  27. Outline • Convex Relaxations • Integer Programming Formulation • LP Relaxation • SDP Relaxation • SOCP Relaxation • Comparing Relaxations • Move Making Algorithms • Some Interesting Open Problems

  28. ∑ xi = 2 - |L| i  va X - xxT 0 Derive SOCP relaxation from the SDP relaxation SOCP Relaxation 1 1 ∑ Pij (1 + xi + xj + Xij) x* = argmin + ∑ ui (1 + xi) 4 2 xi[-1,1] Xii = 1 Further Relaxation

  29. Choose a matrix C1 = UUT 0 Choose a matrix C2 = UUT 0 SOCP Relaxation Kim and Kojima, 2000 Choose a sub-graph G Variables xG and XG C1 (XG - xGxGT) ≥ 0 REPEAT

  30. Outline • Convex Relaxations • Integer Programming Formulation • LP Relaxation • SDP Relaxation • SOCP Relaxation • Comparing Relaxations • Move Making Algorithms • Some Interesting Open Problems

  31. Dominating Relaxation ≥ A B For all MAP Estimation problem (u, P) A dominates B Dominating relaxations are better

  32. Choose a matrix C1 = UUT 0 SOCP Relaxation Kim and Kojima, 2000 Choose a sub-graph G Variables xG and XG C1 (XG - xGxGT) ≥ 0 If G is a tree, LP dominates SOCP

  33. Examples Muramatsu and Suzuki, 2003 (MAXCUT) Ravikumar and Lafferty, 2006 (QP Relaxation) Kumar, Torr and Zisserman, 2006 (Equivalent SOCP Relaxation)

  34. Choose a matrix C1 = UUT 0 SOCP Relaxation Kim and Kojima, 2000 Choose a sub-graph G Variables xG and XG C1 (XG - xGxGT) ≥ 0 If G is a cycle with non-negative P

  35. Choose a matrix C1 = UUT 0 SOCP Relaxation Kim and Kojima, 2000 Choose a sub-graph G Variables xG and XG C1 (XG - xGxGT) ≥ 0 If G is an even cycle with non-positive P

  36. Choose a matrix C1 = UUT 0 SOCP Relaxation Kim and Kojima, 2000 Choose a sub-graph G Variables xG and XG C1 (XG - xGxGT) ≥ 0 If G is an odd cycle with 1 non-positive P

  37. SOCP Relaxation Kumar, Kolmogorov and Torr, 2007 What about other cycles? Dominated by linear cycle inequalities Cliques? Dominated by clique inequalities

  38. Outline • Convex Relaxations • Move Making Algorithms • State of the Art • Comparison with LP Relaxation • Improved Moves • Some Interesting Open Problems

  39. MRFs in Vision lk Pab(i,k) Pab(i,k) = wab min{ d(i-k), M } li wab is non-negative d(.) is a semi-metric distance ub(k) ua(i) vb va Truncated Linear Truncated Quadratic

  40. Current Solution Search Neighbourhood Optimal Move Move Making Energy Solution Space Slide courtesy of Pushmeet Kohli

  41. Outline • Convex Relaxations • Move Making Algorithms • State of the Art • Comparison with LP Relaxation • Improved Moves • Some Interesting Open Problems

  42. Expansion Move Variables take label or retain current label Boykov, Veksler, Zabih 2001 Slide courtesy of Pushmeet Kohli

  43. Expansion Move Variables take label or retain current label Tree Ground House Status: Initialize with Tree Expand Ground Expand House Expand Sky Sky Boykov, Veksler, Zabih 2001 Slide courtesy of Pushmeet Kohli [Boykov, Veksler, Zabih]

  44. Outline • Convex Relaxations • Move Making Algorithms • State of the Art • Comparison with LP Relaxation • Improved Moves • Some Interesting Open Problems

  45. Multiplicative Bounds Expansion Bounds as bad as ICM Bounds 2 2 2 + √2 2M O(√M) 2M O(log h) 2M

  46. Outline • Convex Relaxations • Move Making Algorithms • State of the Art • Comparison with LP Relaxation • Improved Moves • Some Interesting Open Problems

  47. Randomized Rounding yi = (1 + xi)/2 y’i = y0 + y1 + … + yi 0 y’0 y’i y’k y’h = 1 Choose an interval of length L’

  48. Randomized Rounding yi = (1 + xi)/2 y’i = y0 + y1 + … + yi r 0 y’0 y’i y’k y’h = 1 Generate a random number r  (0,1]

  49. Randomized Rounding yi = (1 + xi)/2 y’i = y0 + y1 + … + yi r 0 y’0 y’i y’k y’h = 1 Assign label next to r (if within the interval)

  50. Move Making • Initialize the labeling • Choose interval I of L’ labels • Each variable can • Retain old label • Choose a label from I • Choose best labeling va vb Iterate over intervals Non-submodular move? Submodular overestimation

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