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ICCV2009: MAP Inference in Discrete Models: Part 3

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This document discusses belief propagation for solving MAP inference problems in discrete models. It begins by explaining that belief propagation can find the exact MAP solution for chains and trees. For general graphs, belief propagation provides an approximate MAP solution. The document outlines how belief propagation works by reparameterizing the energy function through message passing along edges in the graph. This reparameterization ensures that the minimum of the min-marginals is equal to the MAP estimate for chains and trees.

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MAP Inference in Discrete Models M. Pawan Kumar, Stanford University
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 !!!
Energy Minimization Problems NP-Hard MAXCUT CSP NP-hard Space of Problems
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]
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]
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, ...)  …
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)
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)
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
Outline • Reparameterization • Belief Propagation • Tree-reweighted Message Passing
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
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; )
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’
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; )
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; )
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
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 ’
Outline • Reparameterization • Belief Propagation – Exact MAP for Chains and Trees – Approximate MAP for general graphs – Computational Issues and Theoretical Properties • Tree-reweighted Message Passing
Belief Propagation • Some MAP problems are easy • Belief Propagation gives exact MAP for chains • Exact MAP for trees • Clever Reparameterization
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
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
Two Variables 2 2 0 4 -2 1 5 -3 5 5 Va Vb Va Vb Choose the right constant ’b;k = qb;k
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
Computational Complexity • Each constant takes O(|L|) • Number of constants - O(|E||L|) O(|E||L|2) • Memory required ? O(|E||L|)
Belief Propagation on Trees Va Vb Vc Vd Ve Vg Vh Forward Pass: Leaf  Root Backward Pass: Root  Leaf All min-marginals are computed
Outline • Reparameterization • Belief Propagation – Exact MAP for Chains and Trees – Approximate MAP for general graphs – Computational Issues and Theoretical Properties • Tree-reweighted Message Passing
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)
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
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
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
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
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
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?
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
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
Belief Propagation • Generalizes to any arbitrary random field • Complexity per iteration ? O(|E||L|2) • Memory required ? O(|E||L|)
Outline • Reparameterization • Belief Propagation – Exact MAP for Chains and Trees – Approximate MAP for general graphs – Computational Issues and Theoretical Properties • Tree-reweighted Message Passing
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
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
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
Summary of BP Exact for chains Exact for trees Approximate MAP for general cases Convergence not guaranteed So can we do something better?
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
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
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
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
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
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
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
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 ….]
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*
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
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
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
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
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
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
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
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
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
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
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
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
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?
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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.
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
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.
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
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
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
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
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.
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
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
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
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
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
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
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
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
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
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
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

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ICCV2009: MAP Inference in Discrete Models: Part 3

  • 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