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 AB = ∑ Aij Bij

yi2 1

(yi + yj)2 tij

(yi - yj)2 zij

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

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 )

- 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 + QY, 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

- S is a set of semidefinite matrices. S = U UT S

( C )

- Choice of S is crucial for accuracy and efficiency.

YS - ||UTy||2 0

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