Solving Markov Random Fields using Second Order Cone Programming
Download
1 / 1

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


  • 98 Views
  • Uploaded on
  • Presentation posted in: General

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.

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

Download Presentation

Solving Markov Random Fields using Second Order Cone Programming

An Image/Link below is provided (as is) to download presentation

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

  • 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


ad
  • Login