Solving Markov Random Fields using Second Order Cone Programming - PowerPoint PPT Presentation

1 / 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.

I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.

Solving Markov Random Fields using Second Order Cone Programming

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.

- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -

Presentation Transcript

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

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*

( 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

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

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