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Focused Inference with Local Primal-Dual Gaps

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1 Focused Inference with Local Primal-Dual Gaps
Dhruv Batra (TTIC) Joint work with: Daniel Tarlow (U Toronto), Sebastian Nowozin (MSRC), Pushmeet Kohli (MSRC), Vladimir Kolmogorov (UCL)

2 Overview Discrete Labelling Problems in Computer Vision
Focused Inference Local Primal-Dual Gap [Batra, Nowozin, Kohli AISTATS ‘11] [Batra, Kohli CVPR ‘11] [Tarlow, Batra, Kohli, Kolmogorov ICML ‘11] (C) Dhruv Batra

3 Labelling Problems in Vision
Segmentation Stereo Left image Right image Disparity map Motion Flow Geometric Labelling Denoising Name-Face Association (C) Dhruv Batra

4 MAP Inference in MRFs Set of discrete random variables
(Pairwise) Cost/Energy Function MAP Inference X1 X2 … Xn Edge Energies / Distributed Prior Node Energies / Local Costs kx1 kxk (C) Dhruv Batra

5 Inference MAP problem In general NP-hard [Shimony ‘94]
Approximate Inference Loopy BP [Pearl, ‘88] α-Expansion [Boykov ’01, Komodakis ‘05] LP Relaxations [Schlesinger ‘76, Wainwright ’05, Sontag ‘08] Outer-Planar & High-order Decompositions [Batra ‘10, Kappes ‘10] (C) Dhruv Batra

6 Approximate Inference
Carpet Bombing (C) Dhruv Batra

7 Focused Inference (C) Dhruv Batra

8 Focused Inference (C) Dhruv Batra Focused Inference
Energy-Aware Message-Passing ICML ‘11 Label Re-ordering in α-Expansion CVPR ‘11 Tightening LP Relaxations AISTATS ‘11 Ordering of Labels Move Number Classical Expansions Our Guided Expansions 1 Airplane Sheep 2 Bicycle Dog 3 Bird 4 Boat Cow 5 Bottle Cat (C) Dhruv Batra

9 Common Theme LP-relaxation
[Schlesinger ‘76, Koster ’98, Chekuri ‘01, Wainwright ’05] Primal LP Computation Objective Dual LP (C) Dhruv Batra

10 Current Primal-Dual Gap
Common Theme LP-relaxation [Schlesinger ‘76, Koster ’98, Chekuri ‘01, Wainwright ’05] Local Primal-Dual Gaps Primal contribution minus Dual Contribution Distributed Primal-Dual Gap Generalization of Complimentary Slackness Conditions Computation Objective Current Primal-Dual Gap (C) Dhruv Batra

11 MAP-MRF Over-Complete Representation kx1 k2x1 (C) Dhruv Batra

12 MAP-MRF Energy kx1 k2x1 (C) Dhruv Batra

13 Consistent Assignments
MAP-MRF Integer Program Indicator Variables Unique Label Consistent Assignments (C) Dhruv Batra

14 Tractable (but not scalable)
LP Relaxation Linear Program Tractable (but not scalable) (C) Dhruv Batra

15 LP Relaxation Linear Program Dual Program (C) Dhruv Batra

16 LP Relaxation Interpretation of Dual Program
Independently minimize terms Subject to Reparameterization (C) Dhruv Batra

17 LP Relaxation Solving LP Block Co-ordinate Ascent on Dual
Choose a block (set) of variables Optimize block; fix rest Repeat (C) Dhruv Batra

18 LP Relaxation Linear Program --- Dual Program Complementary Slackness:
(C) Dhruv Batra

19 Local Primal-Dual Gap Local Primal-Dual Gap
Defined for nodes & edges (higher order extensions later) Contribution of each node and edge to the Primal-Dual Gap Dual Primal (C) Dhruv Batra

20 Local Primal-Dual Gap Intuition (C) Dhruv Batra

21 Local Primal-Dual Gap Properties Decomposability Quickly Computable
Sums to the total Primal-Dual Gap Can easily define LPDG for sub-graphs If no sub-graph with strictly positive LPDG exists, LP is tight (C) Dhruv Batra

22 Focused Inference (C) Dhruv Batra Focused Inference
Energy-Aware Message-Passing ICML ‘11 Label Re-ordering in α-Expansion CVPR ‘11 Tightening LP Relaxations AISTATS ‘11 Ordering of Labels Move Number Classical Expansions Our Guided Expansions 1 Airplane Sheep 2 Bicycle Dog 3 Bird 4 Boat Cow 5 Bottle Cat (C) Dhruv Batra

23 Focused Inference Dynamic Tree Block Coordinate Ascent [ICML ‘11]
Image Current Seg. Update Mask Updated Seg. Messages (C) Dhruv Batra

24 Focused Inference (C) Dhruv Batra Focused Inference
Energy-Aware Message-Passing ICML ‘11 Label Re-ordering in α-Expansion CVPR ‘11 Tightening LP Relaxations AISTATS ‘11 Ordering of Labels Move Number Classical Expansions Our Guided Expansions 1 Airplane Sheep 2 Bicycle Dog 3 Bird 4 Boat Cow 5 Bottle Cat (C) Dhruv Batra

25 Dynamic Re-ordering of Blocks
Goal: Category Segmentation α-Expansion solves the standard LP relaxation Loop over α Current Soln 2-Label Problem + GC New Soln α-Expansion α (C) Dhruv Batra

26 Dynamic Re-ordering of Blocks
Ordering of Labels Move Number Classical Expansions Our Guided Expansions Image 1 1 Airplane Car 2 Bicycle Person 3 Bird Motorbike 4 Boat Train 5 Bottle Image 2 Sheep Dog Cow Cat (C) Dhruv Batra

27 Dynamic Re-ordering of Blocks
LPDG score For each node i, label Dual Primal (C) Dhruv Batra

28 Experiments (C) Dhruv Batra

29 Focused Inference (C) Dhruv Batra Focused Inference
Energy-Aware Message-Passing ICML ‘11 Label Re-ordering in α-Expansion CVPR ‘11 Tightening LP Relaxations AISTATS ‘11 Ordering of Labels Move Number Classical Expansions Our Guided Expansions 1 Airplane Sheep 2 Bicycle Dog 3 Bird 4 Boat Cow 5 Bottle Cat (C) Dhruv Batra

30 LP Relaxation LP-relaxation
[Schlesinger ‘76, Koster ’98, Chekuri ‘01, Wainwright ’05] Primal LP Computation Objective Dual LP (C) Dhruv Batra

31 LP Relaxation LP-relaxation
[Schlesinger ‘76, Koster ’98, Chekuri ‘01, Wainwright ’05] Primal LP Computation Objective Dual LP (C) Dhruv Batra

32 Hierarchy of LPs LPDG to the rescue! Increasingly Complex Sub-problems
Edge-Consistent LP Triplet-Clique Consistent LP LPDG to the rescue! -- Score Clusters / Constraints -- Add high scoring ones (C) Dhruv Batra

33 Experiments (C) Dhruv Batra

34 Summary Focused Inference vs. Energy-Agnostic Inference
Exploiting structure of the problem First wave of success Submodularity Distance Transforms Truncated Convex Potentials Second wave of success Focused Message Passing Task Specific Computation (C) Dhruv Batra

35 Thank You!

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