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

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

Dhruv Batra

(Currently)Research Assistant Professor

TTI-Chicago

(Spring 2013)Assistant Professor

Virginia Tech

Local Ambiguity• Graphical Models

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x1

x2

…

xn

MAP

Inference

Most Likely AssignmentMAP Problem

Cat

Hat

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)

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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

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History

MAP M-Best MAP

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History

MAP M-Best MAP

Message-Passing Algs- Dynamic Programming- Belief Propagation style

[Pearl ’82], [Lauritzen & Spiegelhalter ‘88],

[Shafer & Shenoy ‘86]

[Seroussi & Golmard ‘94], [Flerova & Dechter ‘10], [Yanover & Weiss ‘03], [Flerova & Dechter ’11]

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History

MAP M-Best MAP





Linear ProgrammingFormulation

[Schlesinger ‘76], [Wainwright et al. ‘05],

[Komodakis ’07][Fromer & Globerson ’09]

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History

MAP M-Best MAP





Linear ProgrammingFormulation


[Komodakis ’07][Fromer & Globerson ’09]

Message-Passing for solving LP


[Kolmogorov ‘06], [Komodakis ’07], [Werner ’07]

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This Work[Batra UAI ’12]?

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

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10 32 100 316 1000 3162 10000 31614 1000000

20

40

60

80

100

120

140

160

LP-solver

Our Approach

#Nodes

Tim

e (s

ec)

Better

Outline

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Tree-MRFM=2

Tree-MRFM>2

Loopy MRFM=2

Loopy MRFM>2

M

Cycles

- Partition Enumeration Scheme [Fromer & Globerson NIPS09]

- Others

Details in Paper

M=2 M>2 Schemes

Background

• Over-Complete Representation

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x1

x2

…

xnXikxk

……

…

kx1

…

……

… …

kx1

1

1

0

0

0

0

0

0

1

0

0

0

0

1

0

0

Background

• Over-Complete Representation

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x1

x2

…

xnXi

… …

k2x1

100000000000

010000000000

Background• MAP Integer Program

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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

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Outline

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Tree-MRFM=2

Tree-MRFM>2

Loopy MRFM=2

Loopy MRFM>2

M

Cycles

M-Best MAP LP: Tree

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Spanning-Tree Inequality

[Fromer & Globerson NIPS09]

M-Best MAP LP: Tree

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~ 106 x 106

Generic LP-solver: CPLEX

[Fromer & Globerson NIPS09]

M-Best MAP LP: Tree• Lagrangian Relaxation

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Dualize

2-PassBelief Propagation

Similarity-Augmented Energy


• Dual Problem

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2nd Best MAP energy

Concave (Non-smooth)

Lower-Bound on 2nd Best MAP energy

upergradient Ascent


• Dual Problem

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upergradient Ascent

Primal Block Dual Block

primal point

dual point


• Dual Problem

• Guarantees– Suitable choice of stepsize solves Lagrangian [Shor ‘85]

– LP => Strong Duality

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upergradient Ascent

Outline

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Tree-MRFM=2

Tree-MRFM>2

Loopy MRFM=2

Loopy MRFM>2

M

Cycles

M-Best MAP LP: Loopy-MRFs

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…,,

M-Best MAP LP: Loopy-MRFs

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Dualize

…,,

Problems1. Exponentially many Lagrangian Terms2. Collection of factors not a tree

Exponentially Many Terms

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Primal Block

primal point

Dual Block

dual point

Constraint Management

primal

point

dual point

Tree Subset

upergradient Ascent

Dynamic

…

Exponentially Many Terms

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Primal Block Dual Block

Constraint Management

primal

point

dual point

Tree Subset

…, ,

upergradient Ascent

Dynamic

Max-Weight Spanning TreeSame as [Fromer & Globerson]

Loopy Graph

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Problems1. Exponentially many Lagrangian Terms2. Collection of factors not a tree

…

Dual Decomposition

M-Best MAP LP: Loopy-MRFs• Guarantees

– Dynamic Supergradient Ascent w/ Max-Violation Oracle solves Lagrangian Relaxation [Emiel & Sagastizabal ‘08]

– LP => Strong Duality

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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]

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Results: Tree-MRFs

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Better

Results: Loopy-MRFs

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Better

Extension: Diverse M-Best

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Diverse M-Best Solutions in MRFsBatra, Yadollahpour, Guzman, ShakhnarovichECCV 2012

Task-Specific

Diversity

Extension: Diverse M-Best• Interactive Segmentation

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Image + Scribbles 2nd Best Mode2nd Best MAPMAP

1-2 Nodes Flipped 100-500 Nodes Flipped

Extension: Diverse M-Best

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Input MAP Best Mode

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 MRFs

Batra, Yadollahpour, Guzman, ShakhnarovichECCV 2012

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Thank you!

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Results: Tree-MRFs

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Quality of Solutions: Loopy-MRFs

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Results: Loopy-MRFs

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Applications• What can we do with multiple solutions?

– More choices for “human/expert in the loop”

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– More choices for “human/expert in the loop”– Input to next system in cascade

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Step 1 Step 2 Step 3Top M

hypotheses

Top M

hypotheses


– More choices for “human in the loop”– Rank solutions

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[Carreira and Sminchisescu, CVPR10]

State-of-art segmentation on PASCAL Challenge 2011

~10,000

Documents

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