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M. Pawan Kumar Center for Visual Computing Ecole Centrale Paris

Rounding-based Moves for Metric Labeling. M. Pawan Kumar Center for Visual Computing Ecole Centrale Paris. Metric Labeling. Post. Random variables V = {v 1 , v 2 , …, v n }. Label set L = {l 1 , l 2 , …, l h }. Labeling y ∈ L n. Labelings quantatively distinguished by energy E( y ).

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M. Pawan Kumar Center for Visual Computing Ecole Centrale Paris

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  1. Rounding-based Moves for Metric Labeling M. Pawan Kumar Center for Visual Computing Ecole Centrale Paris

  2. Metric Labeling Post Random variables V = {v1, v2, …, vn} Label set L = {l1, l2, …, lh} Labeling y ∈ Ln Labelingsquantatively distinguished by energy E(y) Unary potential of variable va∈ V • ∑aθa(ya)

  3. Metric Labeling Post Random variables V = {v1, v2, …, vn} Label set L = {l1, l2, …, lh} Labeling y ∈ Ln Labelingsquantatively distinguished by energy E(y) Pairwise potential of variables (va,vb) • miny • ∑aθa(ya) • + ∑(a,b)wab d(ya,yb) wab is non-negative d(.,.) is a metric distance function

  4. Outline Post • Existing Work • Move-Making Algorithms (Efficient) • Linear Programming Relaxation (Accurate) • Rounding-based Moves • Equivalence • Complete Rounding and Complete Move • Interval Rounding and Interval Move • Hierarchical Rounding and Hierarchical Move

  5. Expansion Algorithm Post Variables take label lα or retain current label Tree Ground Initialize with Tree Expand Ground Expand House Expand Sky House Sky Boykov, Veksler and Zabih, ICCV 1999

  6. Move-Making Algorithms Post Start with an initial labeling y0 Iteration t Define St ⊆ Ln containing current labeling yt • ∑aθa(ya) • + ∑(a,b)wab d(ya,yb) • argminy • yt+1 = • s.t. • y ∈ St Above problem is easier than original problem Sometimes it can even be solved exactly

  7. Outline Post • Existing Work • Move-Making Algorithms (Efficient) • Linear Programming Relaxation (Accurate) • Rounding-based Moves • Equivalence • Complete Rounding and Complete Move • Interval Rounding and Interval Move • Hierarchical Rounding and Hierarchical Move

  8. Linear Programming Relaxation Post Binary indicator xa(i) ∈ {0,1} If variable ‘a’ takes the label ‘i’ then xa(i) = 1 Each variable ‘a’ takes one label ∑ixa(i) = 1 Similarly, binary indicator xab(i,k) ∈ {0,1} Chekuri, Khanna, Naor and Zosin, SODA 2001

  9. Linear Programming Relaxation Post Minimize a linear function over feasible x Rounding Relaxed xa(i), xab(i,k)  [0,1] Indicators xa(i), xab(i,k)  {0,1} Chekuri, Khanna, Naor and Zosin, SODA 2001

  10. Outline Post • Existing Work • Move-Making Algorithms (Efficient) • Linear Programming Relaxation (Accurate) • Rounding-based Moves • Equivalence • Complete Rounding and Complete Move • Interval Rounding and Interval Move • Hierarchical Rounding and Hierarchical Move

  11. Move-Making Bound Post y*: Optimal Labeling y: Estimated Labeling Σaθa(ya) + Σ(a,b)wabd(ya,yb) ≥ Σaθa(y*a) + Σ(a,b)wabd(y*a,y*b)

  12. Move-Making Bound Post y*: Optimal Labeling y: Estimated Labeling Σaθa(ya) + Σ(a,b)wabd(ya,yb) ≤ B Σaθa(y*a) + Σ(a,b)wabd(y*a,y*b) For all possible values of θa(i) and wab

  13. Rounding Approximation Post x*: LP Optimal Solution x: Rounded Solution ΣaΣiθa(i)xa(i) + Σ(a,b)Σ(i,k) wabd(i,k)xab(i,k) ≥ ΣaΣiθa(i)x*a(i) + Σ(a,b)Σ(i,k) wabd(i,k)x*ab(i,k)

  14. Rounding Approximation Post x*: LP Optimal Solution x: Rounded Solution ΣaΣiθa(i)xa(i) + Σ(a,b)Σ(i,k) wabd(i,k)xab(i,k) ≤ A ΣaΣiθa(i)x*a(i) + Σ(a,b)Σ(i,k) wabd(i,k)x*ab(i,k) For all possible values of θa(i) and wab

  15. Equivalence Post For any known rounding with approximation A there exists a move-making algorithm such that the move-making bound B = A We know how to design such an algorithm

  16. Outline Post • Existing Work • Move-Making Algorithms (Efficient) • Linear Programming Relaxation (Accurate) • Rounding-based Moves • Equivalence • Complete Rounding and Complete Move • Interval Rounding and Interval Move • Hierarchical Rounding and Hierarchical Move

  17. Complete Rounding Post Treat x*a(i)  [0,1] as probability that ya = li Cumulative probability za(i) = Σj≤ix*a(j) r za(2) za(i) za(k) 0 za(1) za(h) = 1 Generate a random number r  (0,1] Assign the label next to r

  18. Complete Rounding - Example Post 0.25 0.5 0.75 1.0 r za(2) za(3) 0 za(1) za(4) 0.7 0.8 0.9 1.0 r zb(1) zb(3) 0 zb(4) zb(2) 0.2 0.3 0.1 1.0 r 0 zc(3) zc(2) zc(4) zc(1)

  19. Equivalent Move Post Complete Move !!

  20. Complete Move Post Start with an initial labeling y0 Iteration t Define St⊆ Ln • ∑aθa(ya) • + ∑(a,b)wab d(ya,yb) • argminy • yt+1 = • s.t. • y ∈ St

  21. Complete Move Post Start with an initial labeling y0 Iteration t Define St= Ln • ∑aθa(ya) • + ∑(a,b)wab d(ya,yb) • argminy • yt+1 = • s.t. • y ∈ St Above problem is the same as the original problem How do we solve this problem?

  22. Complete Move Post Define St= Ln • ∑aθa(ya) • + ∑(a,b)wab d’(ya,yb) • argminy • yt+1 = • s.t. • y ∈ St Above problem is the same as the original problem How do we solve this problem?

  23. Complete Move Post Define St= Ln • ∑aθa(ya) • + ∑(a,b)wab d’(ya,yb) • argminy • yt+1 = • s.t. • y ∈ St Submodular overestimation d’ of d Obtained by solving a small LP

  24. Submodular Overestimation Post • mind’ • maxi,kd’(li,lk)/d(li,lk) • d’(li,lk) ≥ d(li,lk) • s.t. d’(li,lk+1) + d’(li+1,lk) ≥ d(li,lk) + d(li+1,lk+1)

  25. Submodular Overestimation Post • mind’ • b • d’(li,lk) ≥ d(li,lk) • s.t. d’(li,lk+1) + d’(li+1,lk) ≥ d(li,lk) + d(li+1,lk+1) • bd(li,lk) ≥ d’(li,lk) • Dual provides worst-case instance • for complete rounding

  26. Outline Post • Existing Work • Move-Making Algorithms (Efficient) • Linear Programming Relaxation (Accurate) • Rounding-based Moves • Equivalence • Complete Rounding and Complete Move • Interval Rounding and Interval Move • Hierarchical Rounding and Hierarchical Move

  27. Interval Rounding Post Treat x*a(i)  [0,1] as probability that ya = li Cumulative probability za(i) = Σj≤ix*a(j) za(2) za(i) za(k) 0 za(1) za(h) = 1 Choose an interval of length h’

  28. Interval Rounding Post Treat x*a(i)  [0,1] as probability that ya = li Cumulative probability za(i) = Σj≤ix*a(j) r 0 za(k)-za(i) REPEAT Choose an interval of length h’ Generate a random number r  (0,1] Assign the label next to r if it is within the interval

  29. Interval Rounding - Example Post 0.25 0.5 0.75 1.0 za(2) za(3) 0 za(1) za(4) 0.7 0.8 0.9 1.0 zb(1) zb(3) 0 zb(4) zb(2) 0.2 0.3 0.1 1.0 0 zc(3) zc(2) zc(4) zc(1)

  30. Interval Rounding - Example Post 0.25 0.5 r za(2) 0 za(1) 0.7 0.8 r zb(1) 0 zb(2) 0.2 0.1 r 0 zc(2) zc(1)

  31. Interval Rounding - Example Post 0.25 0.5 0.75 1.0 za(2) za(3) 0 za(1) za(4) 0.7 0.8 0.9 1.0 zb(1) zb(3) 0 zb(4) zb(2) 0.2 0.3 0.1 1.0 0 zc(3) zc(2) zc(4) zc(1)

  32. Interval Rounding - Example Post 0.2 0.3 0.1 1.0 0 zc(3) zc(2) zc(4) zc(1)

  33. Interval Rounding - Example Post 0.1 0.2 r zc(3) zc(2) 0 -zc(1) -zc(1)

  34. Interval Rounding - Example Post 0.25 0.5 0.75 1.0 za(2) za(3) 0 za(1) za(4) 0.7 0.8 0.9 1.0 zb(1) zb(3) 0 zb(4) zb(2) 0.2 0.3 0.1 1.0 0 zc(3) zc(2) zc(4) zc(1)

  35. Equivalent Move Post Interval Move !!

  36. Interval Move Post Start with an initial labeling y0 Iteration t Choose an interval of labels of length h’ y ∈ Stiffya = yta or ya∈ interval of labels • ∑aθa(ya) • + ∑(a,b)wab d(ya,yb) • argminy • yt+1 = • s.t. • y ∈ St How do we solve this problem?

  37. Interval Move Post Start with an initial labeling y0 Iteration t Choose an interval of labels of length h’ y ∈ Stiffya = yta or ya∈ interval of labels • ∑aθa(ya) • + ∑(a,b)wab d’(ya,yb) • argminy • yt+1 = • s.t. • y ∈ St Submodular overestimation d’ of d

  38. Outline Post • Existing Work • Move-Making Algorithms (Efficient) • Linear Programming Relaxation (Accurate) • Rounding-based Moves • Equivalence • Complete Rounding and Complete Move • Interval Rounding and Interval Move • Hierarchical Rounding and Hierarchical Move

  39. Hierarchical Rounding Post L1 L2 L3 l1 l2 l3 l4 l5 l6 l7 l8 l9 Hierarchical clustering of labels (e.g. r-HST metrics)

  40. Hierarchical Rounding Post L1 L2 L3 l1 l2 l3 l4 l5 l6 l7 l8 l9 Assign variables to labels L1, L2 or L3 Move down the hierarchy until the leaf level

  41. Hierarchical Rounding Post L1 L2 L3 l1 l2 l3 l4 l5 l6 l7 l8 l9 Assign variables to labels l1, l2 or l3

  42. Hierarchical Rounding Post L1 L2 L3 l1 l2 l3 l4 l5 l6 l7 l8 l9 Assign variables to labels l4, l5 or l6

  43. Hierarchical Rounding Post L1 L2 L3 l1 l2 l3 l4 l5 l6 l7 l8 l9 Assign variables to labels l7, l8 or l9

  44. Equivalent Move Post Hierarchical Move !!

  45. Hierarchical Move Post L1 L2 L3 l1 l2 l3 l4 l5 l6 l7 l8 l9 Hierarchical clustering of labels (e.g. r-HST metrics)

  46. Hierarchical Move Post L1 L2 L3 l1 l2 l3 l4 l5 l6 l7 l8 l9 Obtain labeling y1 restricted to labels {l1,l2,l3}

  47. Hierarchical Move Post L1 L2 L3 l1 l2 l3 l4 l5 l6 l7 l8 l9 Obtain labeling y2 restricted to labels {l4,l5,l6}

  48. Hierarchical Move Post L1 L2 L3 l1 l2 l3 l4 l5 l6 l7 l8 l9 Obtain labeling y3 restricted to labels {l7,l8,l9}

  49. Hierarchical Move Post L1 L2 L3 y3(a) y3(b) y2(a) y2(b) y1(a) y1(b) Va Vb Move up the hierarchy until we reach the root

  50. Questions? http://mpawankumar.info

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