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Partial Optimality by Pruning for MAP-inference with General - - PowerPoint PPT Presentation

Mar 08, 2023 90 likes •499 views

Partial Optimality by Pruning for MAP-inference with General Graphical Models Paul Swoboda, Bogdan Savchynskyy, J org Kappes, Christoph Schn orr Heidelberg University, Germany 1/1 Segment the image... 2/1 Optimal labeling 3/1

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SLIDE 1

Partial Optimality by Pruning for MAP-inference with General Graphical Models

Paul Swoboda, Bogdan Savchynskyy, J¨

  • rg Kappes,

Christoph Schn¨

  • rr

Heidelberg University, Germany

1/1

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SLIDE 2

Segment the image...

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SLIDE 3

Optimal labeling

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SLIDE 4

Optimal labeling NP hard

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SLIDE 5

Optimal labeling NP hard Solve convex relaxation (LP) Round relaxed solution NO optimality guarantees

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SLIDE 6

Optimal labeling NP hard Partial labeling Polynomially solvable Optimality guaranteed

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SLIDE 7

Energy Minimization - MAP-Inference with Graphical Models

arg minxPXJpxq :“ ÿ

fPF

θf pxnepfqq

  • variable

xi P t1, . . . , Nu

  • factor

f P F Ă Vn θf pxnepfqq

  • potential of xnepfq P t1, . . . , Nu|nepfq|

Factor graph G “ pV, F, Eq

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SLIDE 8

Easy Examples

Figure : Color segmentation [Lellman 2010]

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SLIDE 9

Difficult Examples

(a) color segmentation (b) stereo (c) panorama stitching

Figure : [Lellman 2010],[Szeliski et al. 2008],[Agarwala et al. 2004]

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SLIDE 10

Related Work

Work non-binary higher order non-Potts

  • Aux. problem

Boros & Hammer 2002 ´ ´ ` QPBO Kovtun 2003 ` ´ ´ submodular Rother et al. 2007 ´ ´ ` QPBO Kohli et al. 2008 ` ´ ` QPBO Kovtun 2005 ` ´ ` submodular Fix et al. 2011 ´ ` ` QPBO Kahl & Strandmark 2012 ´ ` ` bi-submodular Windheuser et al. 2012 ` ` ` bi-submodular Swoboda et al. 2013 ` ´ ´ LP Shekhovtsov 2014 ` ´ ` LP Ours ` ` ` any relaxation

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SLIDE 11

Related Work

Work non-binary higher order non-Potts

  • Aux. problem

Boros & Hammer 2002 ´ ´ ` QPBO Kovtun 2003 ` ´ ´ submodular Rother et al. 2007 ´ ´ ` QPBO Kohli et al. 2008 ` ´ ` QPBO Kovtun 2005 ` ´ ` submodular Fix et al. 2011 ´ ` ` QPBO Kahl & Strandmark 2012 ´ ` ` bi-submodular Windheuser et al. 2012 ` ` ` bi-submodular Swoboda et al. 2013 ` ´ ´ LP Shekhovtsov 2014 ` ´ ` LP Ours ` ` ` any relaxation

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SLIDE 12

Related Work

Work non-binary higher order non-Potts

  • Aux. problem

Boros & Hammer 2002 ´ ´ ` QPBO Kovtun 2003 ` ´ ´ submodular Rother et al. 2007 ´ ´ ` QPBO Kohli et al. 2008 ` ´ ` QPBO Kovtun 2005 ` ´ ` submodular Fix et al. 2011 ´ ` ` QPBO Kahl & Strandmark 2012 ´ ` ` bi-submodular Windheuser et al. 2012 ` ` ` bi-submodular Swoboda et al. 2013 ` ´ ´ LP Shekhovtsov 2014 ` ´ ` LP Ours ` ` ` any relaxation

7/1

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SLIDE 13

Related Work

Work non-binary higher order non-Potts

  • Aux. problem

Boros & Hammer 2002 ´ ´ ` QPBO Kovtun 2003 ` ´ ´ submodular Rother et al. 2007 ´ ´ ` QPBO Kohli et al. 2008 ` ´ ` QPBO Kovtun 2005 ` ´ ` submodular Fix et al. 2011 ´ ` ` QPBO Kahl & Strandmark 2012 ´ ` ` bi-submodular Windheuser et al. 2012 ` ` ` bi-submodular Swoboda et al. 2013 ` ´ ´ LP Shekhovtsov 2014 ` ´ ` LP Ours ` ` ` any relaxation

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SLIDE 14

Algorithm Outline

Initialize: Generate labeling proposal repeat Verify the proposal on a current graph Shrink the graph until verification succeeds

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SLIDE 15

Algorithm Outline

Initialize: Generate labeling proposal repeat Verify the proposal on a current graph Shrink the graph until verification succeeds

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SLIDE 16

Algorithm Outline

Initialize: Generate labeling proposal repeat Verify the proposal on a current graph Shrink the graph until verification succeeds

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SLIDE 17

Algorithm Outline

Initialize: Generate labeling proposal repeat Verify the proposal on a current graph Shrink the graph until verification succeeds

8/1

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SLIDE 18

Algorithm

Proposed Partial Labeling

Jpxq “ ÿ

vPV

θvpxvq ` ÿ

uvPE

θuvpxv, xuq Labeling: x P X Partial optimality: χ P t0, 1u|V|

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SLIDE 19

Algorithm

Proposed Partial Labeling

Jpxq “ ÿ

vPV

θvpxvq ` ÿ

uvPE

θuvpxv, xuq Labeling: x P X Partial optimality: χ P t0, 1u|V|

Perturbed Problem

Jχpxq “ ÿ

vPVYV

θvpxvq` ÿ

uvPE

θuvpxv, xuq` ÿ

vPV

¯ θvpxvq

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SLIDE 20

Algorithm

Proposed Partial Labeling

Jpxq “ ÿ

vPV

θvpxvq ` ÿ

uvPE

θuvpxv, xuq Labeling: x P X Partial optimality: χ P t0, 1u|V|

Perturbed Problem

Jχpxq “ ÿ

vPVYV

θvpxvq` ÿ

uvPE

θuvpxv, xuq` ÿ

vPV

¯ θvpxvq ˆ x “ arg minxPXJχpxq

NP-hard Ñ Relaxation 9/1

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SLIDE 21

Algorithm

Proposed Partial Labeling

Jpxq “ ÿ

vPV

θvpxvq ` ÿ

uvPE

θuvpxv, xuq Labeling: x P X Partial optimality: χ P t0, 1u|V|

Perturbed Problem

Jχpxq “ ÿ

vPVYV

θvpxvq` ÿ

uvPE

θuvpxv, xuq` ÿ

vPV

¯ θvpxvq ˆ x “ arg minxPXJχpxq

NP-hard Ñ Relaxation

ˆ xi ‰ xi Ñ χi “ 0

Shrinking Rule 9/1

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SLIDE 22

Algorithm

Proposed Partial Labeling

Jpxq “ ÿ

vPV

θvpxvq ` ÿ

uvPE

θuvpxv, xuq Labeling: x P X Partial optimality: χ P t0, 1u|V|

Perturbed Problem

Jχpxq “ ÿ

vPVYV

θvpxvq` ÿ

uvPE

θuvpxv, xuq` ÿ

vPV

¯ θvpxvq ˆ x “ arg minxPXJχpxq

NP-hard Ñ Relaxation

ˆ xi ‰ xi Ñ χi “ 0

Shrinking Rule 9/1

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SLIDE 23

Algorithm

Proposed Partial Labeling

Jpxq “ ÿ

vPV

θvpxvq ` ÿ

uvPE

θuvpxv, xuq Labeling: x P X Partial optimality: χ P t0, 1u|V|

Perturbed Problem

Jχpxq “ ÿ

vPVYV

θvpxvq` ÿ

uvPE

θuvpxv, xuq` ÿ

vPV

¯ θvpxvq ˆ x “ arg minxPXJχpxq

NP-hard Ñ Relaxation

ˆ xi ‰ xi Ñ χi “ 0

Shrinking Rule 9/1

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SLIDE 24

Algorithm

Proposed Partial Labeling

Jpxq “ ÿ

vPV

θvpxvq ` ÿ

uvPE

θuvpxv, xuq Labeling: x P X Partial optimality: χ P t0, 1u|V|

Perturbed Problem

Jχpxq “ ÿ

vPVYV

θvpxvq` ÿ

uvPE

θuvpxv, xuq` ÿ

vPV

¯ θvpxvq ˆ x “ arg minxPXJχpxq

NP-hard Ñ Relaxation

ˆ xi ‰ xi Ñ χi “ 0

Shrinking Rule 9/1

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SLIDE 25

Algorithm

Proposed Partial Labeling

Jpxq “ ÿ

vPV

θvpxvq ` ÿ

uvPE

θuvpxv, xuq Labeling: x P X Partial optimality: χ P t0, 1u|V|

Perturbed Problem

Jχpxq “ ÿ

vPVYV

θvpxvq` ÿ

uvPE

θuvpxv, xuq` ÿ

vPV

¯ θvpxvq ˆ x “ arg minxPXJχpxq

NP-hard Ñ Relaxation

ˆ xi ‰ xi Ñ χi “ 0

Shrinking Rule 9/1

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SLIDE 26

Algorithm

Proposed Partial Labeling

Jpxq “ ÿ

vPV

θvpxvq ` ÿ

uvPE

θuvpxv, xuq Labeling: x P X Partial optimality: χ P t0, 1u|V|

Perturbed Problem

Jχpxq “ ÿ

vPVYV

θvpxvq` ÿ

uvPE

θuvpxv, xuq` ÿ

vPV

¯ θvpxvq ˆ x “ arg minxPXJχpxq

NP-hard Ñ Relaxation

ˆ xi ‰ xi Ñ χi “ 0

Shrinking Rule 9/1

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SLIDE 27

Algorithm

Proposed Partial Labeling

Jpxq “ ÿ

vPV

θvpxvq ` ÿ

uvPE

θuvpxv, xuq Labeling: x P X Partial optimality: χ P t0, 1u|V|

Perturbed Problem

Jχpxq “ ÿ

vPVYV

θvpxvq` ÿ

uvPE

θuvpxv, xuq` ÿ

vPV

¯ θvpxvq ˆ x “ arg minxPXJχpxq

NP-hard Ñ Relaxation

ˆ xi ‰ xi Ñ χi “ 0

Shrinking Rule 9/1

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SLIDE 28

Use of Approximate Solvers

Do we need to solve the relaxed problem

ˆ x “ arg minxPXJχpxq

exactly? NO! Approximate solvers with optimality certificate (like TRW-S [Kolmogorov 2005]) are allowed here

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SLIDE 29

Results

Experiment (N) MQPBO Kovtun GRD Fix Ours teddy : : : 0.4423 venus : : : 0.0009 family 0.0432 : : : 0.0611 pano 0.1247 : : : 0.5680 Potts (12) 0.1839 0.7475 : : 0.9231 side-chain (21) 0.0247 : : : 0.6513 protein (8) : : 0.2603 0.2545 0.7799 cell-tracking : : : 0.1771 0.9992 geo-surf (50) : : : : 0.8407

Table : Percentage of persistent variables; : - method inapplicable. We used local polytope relaxation and TRW-S and CPLEX as solvers.

Benchmarks: [Szeliski et al. 2008],[Kappes et al. 2013], [PIC 2011]

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SLIDE 30

Results

Experiment (N) MQPBO Kovtun GRD Fix Ours teddy : : : 0.4423 venus : : : 0.0009 family 0.0432 : : : 0.0611 pano 0.1247 : : : 0.5680 Potts (12) 0.1839 0.7475 : : 0.9231 side-chain (21) 0.0247 : : : 0.6513 protein (8) : : 0.2603 0.2545 0.7799 cell-tracking : : : 0.1771 0.9992 geo-surf (50) : : : : 0.8407

Table : Percentage of persistent variables; : - method inapplicable. We used local polytope relaxation and TRW-S and CPLEX as solvers.

Benchmarks: [Szeliski et al. 2008],[Kappes et al. 2013], [PIC 2011]

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SLIDE 31

Potts models: Results

Ours Kovtun’s method

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SLIDE 32

Potts models: Results

Ours Kovtun’s method

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SLIDE 33

Potts models: Results

Ours Kovtun’s method

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SLIDE 34

Potts models: Results

Ours Kovtun’s method

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SLIDE 35

Potts models: Results

Ours Kovtun’s method

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Take Home Message and Outlook

We presented: A generic method for partial optimality from MAP-Inference, which can employ any relaxation can use certain approximate solvers in the loop (e.g. TRW-S) scales as well as the used MAP-inference solver

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SLIDE 37

Take Home Message and Outlook

We presented: A generic method for partial optimality from MAP-Inference, which can employ any relaxation can use certain approximate solvers in the loop (e.g. TRW-S) scales as well as the used MAP-inference solver Code preliminary research code at http://paulswoboda.net revised code will be included to OpenGM library soon.

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SLIDE 38

Take Home Message and Outlook

We presented: A generic method for partial optimality from MAP-Inference, which can employ any relaxation can use certain approximate solvers in the loop (e.g. TRW-S) scales as well as the used MAP-inference solver Code preliminary research code at http://paulswoboda.net revised code will be included to OpenGM library soon. Future work benefit on research on convex relaxations (tightness and speed) use finer partial optimality criteria (on label level)

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SLIDE 39

Take Home Message and Outlook

We presented: A generic method for partial optimality from MAP-Inference, which can employ any relaxation can use certain approximate solvers in the loop (e.g. TRW-S) scales as well as the used MAP-inference solver Code preliminary research code at http://paulswoboda.net revised code will be included to OpenGM library soon. Future work benefit on research on convex relaxations (tightness and speed) use finer partial optimality criteria (on label level)

Visit our poster number O-2B-5

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