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MAP Estimation in Binary MRFs using Bipartite Multi-Cuts

MAP Estimation in Binary MRFs using Bipartite Multi-Cuts. Sashank J. Reddi Sunita Sarawagi Sundar Vishwanathan Indian Institute of Technology, Bombay. TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: A A A A A A A A A. MAP Estimation. θ i (x i ).

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MAP Estimation in Binary MRFs using Bipartite Multi-Cuts

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  1. MAP Estimation in Binary MRFs using Bipartite Multi-Cuts Sashank J. Reddi Sunita Sarawagi SundarVishwanathan Indian Institute of Technology, Bombay TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AAAAAAAAA

  2. MAP Estimation θi(xi) • Energy Function • MAP Estimation: Find the labeling which minimizes the energy function • NP Hard in general xi θij(xi,xj) xj

  3. Popular Approximation • Based on this LP relaxation • Can be solved efficiently using Message Passing algorithms (TRW-S) and Graph cut based algorithms (QPBO)

  4. Tightening Pairwise LP Relaxation • MPLP (David Sontag et.al 2008), Cycle Repairing Algorithm (Nikos Komodakis et.al 2008) • MPLP uses higher order relaxation

  5. Tightening Pairwise LP Relaxation • MPLP (David Sontag et.al 2008), Cycle Repairing Algorithm (Nikos Komodakis et.al 2008) • MPLP uses higher order relaxation

  6. MAP Estimation via Graph Cuts • Construct a specialized graph for the particular energy function • Minimum cut on this graph minimizes the energy function • Exact MAP in polynomial time when the energy function is sub-modular

  7. Assumption on Parameters • Symmetric, that isand • Zero-normalized, that is and • Any Energy function can be transformed in to an equivalent energy function of this form

  8. Graph Construction 00 01

  9. Graph Construction 00 i0 i1 01

  10. Graph Construction 00 θi(1) i1 i0 θi(0) 01

  11. Graph Construction 00 θi(1) θj(1) θij/2 i1 i0 j0 j1 θi(0) θj(0) 01 θij= θij(0,1)+ θij(1,0)- θij(0,0)- θij(1,1) > 0

  12. Graph Construction 00 θi(1) θj(1) -θij/2 i1 i0 j1 j0 θi(0) θj(0) 01 θij= θij(0,1)+ θij(1,0)- θij(0,0)- θij(1,1) < 0

  13. Bipartite Multi-Cut problem • Given an undirected graph with non-negative edge weights and k ST pairs • Objective: Find the minimum cut with divides the graph into 2 regions and separates the ST pairs • LP and SDP Relaxations (SreyashKenkreet.al 2006) • LP Relaxation gives approximation • SDP Relaxation gives approximation • Bipartite Multi-Cut vs Multi-Cut • Additional constraint on the number of regions the graph is cut

  14. BMC LP • are the ST pairs in the Bipartite Multi-Cut problem • Terminals =

  15. BMC LP is jt de • are the ST pairs in the Bipartite Multi-Cut problem • Terminals = D00(is) D00(jt) 00 01 Dk0(jt) Dk0(is) k1 k0

  16. BMC LP • are the ST pairs in the Bipartite Multi-Cut problem • Terminals = i0 i1 j1 j0

  17. BMC LP is jt de • are the ST pairs in the Bipartite Multi-Cut problem • Terminals = D00(is) D00(jt) 00 01

  18. BMC LP • Infeasible to solve it using LP solvers • Large number of constraints • Combinatorial Algorithm • Solving LP for general graphs may not be easy • Flexibility in construction of the graph • Combinatorial algorithm for a special kind of construction

  19. Symmetric Graph Construction 00 00 θi(1)/2 θj(1)/2 θi(0)/2 θj(0)/2 θij/4 θij/4 i0 j0 i1 j1 θi(0)/2 θj(0)/2 θi(1)/2 θj(1)/2 01 01 θij= θij(0,1)+ θij(1,0)- θij(0,0)- θij(1,1)

  20. Symmetric Graph Construction • More ST pairs are added • Trade off is combinatorial algorithm

  21. Approach of the Algorithm

  22. Multi-Cut LP • denotes all paths between pair of vertices in ST • denotes the set of paths in which contain edge e • Can be solved approximately i.eε-approximation for any error parameter ε Multi – Cut LP Multi -Commodity Flow LP

  23. Symmetric BMC LP (BMC-Sym LP) • Very similar to Multi-Cut LP except for the symmetric constraints

  24. Equivalence of LPs Theorem – When the constructed graph is symmetric, BMC LP, BMC-Sym LP and Multi-Cut LP are equivalent Proof Outline • Any feasible solution of each of the LP can be transformed into a feasible solution of other LPs without changing the objective value

  25. Combinatorial Algorithm Primal Step Dual Step Let Update the flow in the path by Update flow in complementary path • Find shortest path P between • Update • Update Complementary path • Converges in • Can be improved to (Fleischer 1999)

  26. Relationship with Cycle Inequalities • BMC LP is closely related to cycle inequalities • BMC LP with slight modification is equivalent to cycle inequalities • Relates our work to many recent works solving cycle inequalities

  27. Empirical Results • Data Sets • Synthetic Problems • Clique Graphical models • Benchmark Data Set • Algorithms • TRW-S • BMC • ε = 0.02 • MPLP • 1000 iterations or up to convergence

  28. Empirical Results

  29. Empirical Results Convergence Comparison of BMC and MPLP

  30. Empirical Results

  31. Conclusion & Future Work • Conclusion • MAP estimation can be reduced to Bipartite Multi-Cut problem • Algorithms for multi commodity flow can be used for MAP estimation • BMC LP is closely related to cycle inequalities • Future Work • Extensions to multi-label graphical models • Combinatorial algorithm for solving SDP relaxation

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