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Solving Markov Random Fields using Second Order Cone Programming

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Aim: To obtain accurate MAP estimate of. Markov Random Fields. Markov Random Field (MRF) ... MAP - accurate. Complexity - high. Y - y yT 0. Objective function ... – PowerPoint PPT presentation

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Title: Solving Markov Random Fields using Second Order Cone Programming


1
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 ei eiT
S (ei - ej) (ei - ej) T
1
2
1, -1 -1, 1
3
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) - ?ij Pij zij
  • Objective function

arg min yT (4l 2P1) P ? Y, Y y yT 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 u2 ? st
Triangular Inequalities
Additional constraints for better accuracy.
Texture
min yT f subject to Ai y bi ? yT ci 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 )
yT q Q ?Y, Y y yT
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 ? -1,1nxn
Prior 0.5 0.5 0.3 0.3 0.5
RTPM Examples
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(yi,yj) lt 0
  • More efficient and less accurate than SDP.
  • Labels for sites i and j should be compatible

?ij P(yi,yj) zij gt 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 - UT y2 ? 0
Code available at http//cms.brookes.ac.uk/staff/P
awanMudigonda/MRFSOCP.zip
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