Solving Markov Random Fields using Second Order Cone Programming
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Solving Markov Random Fields using Second Order Cone Programming. Aim: To obtain accurate MAP estimate of Markov Random Fields. Results. Solving MRFs using SOCP Relaxations. Choice of S. Subgraph Matching. Desirable to eliminate Y (which squares #variables) by using slack variables.

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Solving Markov Random Fields using Second Order Cone Programming

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Solving markov random fields using second order cone programming

Solving Markov Random Fields using Second Order Cone Programming

Aim:To obtain accurate MAP estimate of

Markov Random Fields

Results

Solving MRFs using SOCP Relaxations

Choice of S

Subgraph Matching

  • Desirable to eliminate Y (which squares #variables) by using slack variables.

  • Let ei =[ 0 0 0 … 1 … 0 0 0 ]T

Markov Random Field (MRF)

S = (ei + ej) (ei + ej) T

S = eieiT

S = (ei - ej) (ei - ej) T

1

2

3

[ 1, -1 ; -1, 1]

Configuration Vector y

4

G2 = (V2,E2)

G1 = (V1,E1)

MRF

5

7

[ 5, 2 ; 7, 1]

Likelihood Vector l

  • 1000 synthetic pairs of graphs

  • 5% noise added

Prior Matrix P

MRF Example

#sites S = 2

#labels L = 2

Let AB = ∑ Aij Bij

yi2 1

(yi + yj)2 tij

(yi - yj)2 zij

y* = arg min yT (4l + 2P1) - ∑ijPij zij

  • Objective function

arg min yT (4l + 2P1) + P  Y, Y = yyT

subject to ∑ y(site i) = 2 - L

tij + zij = 4

  • Bound on slack variables tij and zij

MAP y*

Advantages

( A )

  • Converge is guaranteed.

  • No restrictions on the MRF.

Second Order Cone Programming (SOCP)

Object Recognition

  • Fewer variables, faster than SDP.

  • Efficient interior-point algorithms.

Outline

Second Order Cone

|| u ||  t OR ||u||2  st

Triangular Inequalities

Additional constraints for better accuracy.

Texture

min yT f

subject to || Aiy + bi || yTci + di

SOCP

  • At least two of yi, yj and yk have the same sign.

Yij + Yjk + Yik -1

Part likelihood

Spatial Prior

x2 + y2 = z2

  • Constraints can be specified without using Y.

zij + zjk + zik 8

LBP

( B )

yTq + QY, Y = yyT

Convex Relaxations

  • Random subset of inequalities used for efficiency.

Robust Truncated Prior Model

Truncated for incompatible labels.

GBP

Semidefinite (SDP)

Lift and Project (LP)

  • By changing values of prior, P can be made sparse.

  • Max-k-cut

  • MAP - accurate

  • Complexity - high

  • TRW-S, -expansion

  • MAP - inaccurate

  • Complexity - low

SOCP

Reparametrization

Y - y yT  0

Y [-1,1]nxn

RTPM Examples

Prior [0.5 0.5 0.3 0.3 0.5]

Prior [0 0 -0.2 -0.2 0]

Second Order Cone Programming (SOCP)

ROC Curves for 450 +ve and 2400 -ve images

Additional Compatibility Constraints

P(y*i,y*j) < 0

  • More efficient and less accurate than SDP.

  • Labels for sites i and j should be compatible

∑ij P(yi,yj) zij > 0

  • Relaxation

  • S is a set of semidefinite matrices. S = U UT  S

( C )

  • Choice of S is crucial for accuracy and efficiency.

YS - ||UTy||2  0

Code available at http://cms.brookes.ac.uk/staff/PawanMudigonda/MRFSOCP.zip


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