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An Efficient Message-Passing Algorithm for the M-Best MAP Problem

This paper presents a novel message-passing algorithm designed to solve the M-Best MAP (Maximum A Posteriori) problem efficiently. The proposed method demonstrates guaranteed exact solutions while significantly outperforming traditional LP (Linear Programming) solvers in terms of speed. The algorithm can generate multiple hypotheses useful for applications where models exhibit ambiguity, allowing for greater generalization and user interaction. Our findings highlight its applications in various graphical models and the significant improvements over existing approaches like those by Fromer & Globerson.

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An Efficient Message-Passing Algorithm for the M-Best MAP Problem

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  1. An Efficient Message-Passing Algorithm for the M-Best MAP Problem Dhruv Batra • (Currently) • Research Assistant ProfessorTTI-Chicago • (Spring 2013) • Assistant ProfessorVirginia Tech

  2. Local Ambiguity • Graphical Models Hat x1 x2 MAP Inference … xn Cat Most Likely AssignmentMAP Problem (C) Dhruv Batra

  3. Global Ambiguity • “While hunting in Africa, I shot an elephant in my pajamas. How an elephant got into my pajamas, I’ll never know!” • Groucho Marx (1930) (C) Dhruv Batra

  4. M-Best MAP • Useful for: • Generating multiple hypotheses when model is inaccurate • Passing on hypotheses to next stage in cascade • Show multiple solutions to users • Generalization of MAP, thus NP-Hard (C) Dhruv Batra

  5. History (C) Dhruv Batra

  6. History (C) Dhruv Batra

  7. History (C) Dhruv Batra

  8. History • This Work[Batra UAI ’12] ? (C) Dhruv Batra

  9. Contributions • First message-passing alg for solving M-Best MAP LP of [Fromer & Globerson NIPS09] • Guaranteed to get exact solution to LP • Orders of magnitude faster than a generic LP solver LP-solver Time (sec) Our Approach Better #Nodes (C) Dhruv Batra

  10. Outline M M=2  M>2 Schemes • Partition Enumeration Scheme [Fromer & Globerson NIPS09] • Others • Details in Paper Cycles (C) Dhruv Batra

  11. Background x1 • Over-Complete Representation x2 … … kx1 0 0 0 0 1 1 0 0 … kxk Xi xn … … … … … 0 1 0 0 1 0 0 0 … kx1 (C) Dhruv Batra

  12. Background x1 • Over-Complete Representation x2 … Xi xn … … 100000000000 010000000000 (C) Dhruv Batra k2x1

  13. Background • MAP Integer Program (C) Dhruv Batra

  14. Background • MAP Linear Program • Properties • If LP-opt is integral, MAP is found • LP always integral for trees • Efficient message-passing schemes for solving LP (C) Dhruv Batra

  15. Outline M Cycles (C) Dhruv Batra

  16. M-Best MAP LP: Tree Spanning-Tree Inequality [Fromer & Globerson NIPS09] (C) Dhruv Batra

  17. M-Best MAP LP: Tree Generic LP-solver: CPLEX [Fromer & Globerson NIPS09] ~ 106x 106 (C) Dhruv Batra

  18. M-Best MAP LP: Tree Similarity-Augmented Energy • Lagrangian Relaxation Dualize 2-PassBelief Propagation (C) Dhruv Batra

  19. M-Best MAP LP: Tree • Lagrangian Relaxation • Dual Problem upergradient Ascent 2nd Best MAP energy Concave (Non-smooth) Lower-Bound on 2nd Best MAP energy (C) Dhruv Batra

  20. M-Best MAP LP: Tree • Lagrangian Relaxation • Dual Problem upergradient Ascent primal point Primal Block Dual Block dual point (C) Dhruv Batra

  21. M-Best MAP LP: Tree • Lagrangian Relaxation • Dual Problem • Guarantees • Suitable choice of stepsize solves Lagrangian[Shor ‘85] • LP => StrongDuality upergradient Ascent (C) Dhruv Batra

  22. Outline M Cycles (C) Dhruv Batra

  23. M-Best MAP LP: Loopy-MRFs , , … (C) Dhruv Batra

  24. M-Best MAP LP: Loopy-MRFs Dualize Problems 1. Exponentially many Lagrangian Terms 2. Collection of factors not a tree , , … (C) Dhruv Batra

  25. Exponentially Many Terms Dynamic Constraint Management upergradient Ascent Tree Subset primal point primal point Primal Block Dual Block … dual point dual point (C) Dhruv Batra

  26. Exponentially Many Terms Dynamic Constraint Management upergradient Ascent Tree Subset primal point Max-Weight Spanning TreeSame as [Fromer & Globerson] Primal Block Dual Block , … dual point , (C) Dhruv Batra

  27. Loopy Graph Problems 1. Exponentially many Lagrangian Terms 2. Collection of factors not a tree Dual Decomposition … (C) Dhruv Batra

  28. M-Best MAP LP: Loopy-MRFs • Guarantees • Dynamic Supergradient Ascent w/ Max-Violation Oracle solves Lagrangian Relaxation [Emiel & Sagastizabal ‘08] • LP => Strong Duality (C) Dhruv Batra

  29. Experiments • Synthetic Data • Trees • Grid Graphs • Energies sampled from Gaussians • Methods • STEELARS: Spanning TREE LAgrangian Relaxation Scheme [Proposed] • STRIPES [Fromer & Globerson NIPS09] • BMMF [Yanover & Weiss NIPS03] • NILSSON [Nilsson Stat. & Comp. 98] (C) Dhruv Batra

  30. Results: Tree-MRFs Better (C) Dhruv Batra

  31. Results: Loopy-MRFs Better (C) Dhruv Batra

  32. Extension: Diverse M-Best Task-Specific Diversity • Diverse M-Best Solutions in MRFsBatra, Yadollahpour, Guzman, ShakhnarovichECCV 2012 (C) Dhruv Batra

  33. Extension: Diverse M-Best • Interactive Segmentation Image + Scribbles 2nd Best Mode MAP 2nd Best MAP 1-2 Nodes Flipped 100-500 Nodes Flipped (C) Dhruv Batra

  34. Extension: Diverse M-Best Input MAP Best Mode (C) Dhruv Batra

  35. Conclusions • First message-passing alg for solving M-Best MAP LP • Guaranteed to get exact solution to LP • Orders of magnitude faster than a generic LP solver • Extension: • Diverse M-Best Solutions in MRFsBatra, Yadollahpour, Guzman, ShakhnarovichECCV 2012 (C) Dhruv Batra

  36. Thank you! (C) Dhruv Batra

  37. Results: Tree-MRFs (C) Dhruv Batra

  38. Quality of Solutions: Loopy-MRFs (C) Dhruv Batra

  39. Results: Loopy-MRFs (C) Dhruv Batra

  40. Applications • What can we do with multiple solutions? • More choices for “human/expert in the loop” (C) Dhruv Batra

  41. Applications • What can we do with multiple solutions? • More choices for “human/expert in the loop” • Input to next system in cascade Top M Top M Step 1 Step 2 Step 3 hypotheses hypotheses (C) Dhruv Batra

  42. Applications • What can we do with multiple solutions? • More choices for “human in the loop” • Rank solutions ~10,000 [Carreira and Sminchisescu, CVPR10] State-of-art segmentation on PASCAL Challenge 2011 (C) Dhruv Batra

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