Discrete Optimization Lecture 3 – Part 1 M. Pawan Kumar [email protected] Slides available online

Discrete OptimizationLecture 3 – Part 1

M. Pawan Kumar

[email protected]

Slides available online http://cvn.ecp.fr/personnel/pawan/

Energy Minimization

Va Vb Vc Vd

2

5

4

2

6

3

3

7

0

1 1

0

0

2

1

1

4 1

0

3

Q(f; ) = ∑a a;f(a) + ∑(a,b) ab;f(a)f(b)

Label l0

Label l1

Energy Minimization

Va Vb Vc Vd

2

5

4

2

6

3

3

7

0

1 1

0

0

2

1

1

4 1

0

3

Q(f; ) = ∑a a;f(a) + ∑(a,b) ab;f(a)f(b)

2 + 1 + 2 + 1 + 3 + 1 + 3 = 13

Label l0

Label l1

Energy Minimization

Va Vb Vc Vd

2

5

4

2

6

3

3

7

0

1 1

0

0

2

1

1

4 1

0

3

Q(f; ) = ∑a a;f(a) + ∑(a,b) ab;f(a)f(b)

Label l0

Label l1

Energy Minimization

Va Vb Vc Vd

2

5

4

2

6

3

3

7

0

1 1

0

0

2

1

1

4 1

0

3

Q(f; ) = ∑a a;f(a) + ∑(a,b) ab;f(a)f(b)

5 + 1 + 4 + 0 + 6 + 4 + 7 = 27

Label l0

Label l1

Energy Minimization

Va Vb Vc Vd

2

5

4

2

6

3

3

7

0

1 1

0

0

2

1

1

4 1

0

3

Q(f; ) = ∑a a;f(a) + ∑(a,b) ab;f(a)f(b)

f* = arg min Q(f; )

q* = min Q(f; ) = Q(f*; )

Label l0

Label l1

Min-Marginals

Va Vb Vc Vd

2

5

4

2

6

3

3

7

0

1 1

0

0

2

1

1

4 1

0

3

f* = arg min Q(f; ) such that f(a) = i

Min-marginal qa;i

Label l0

Label l1

Min-Marginals and MAP• Minimum min-marginal of any variable = energy of MAP labelling

minf Q(f; ) such that f(a) = i

qa;i mini

mini ( )

Va has to take one label

minf Q(f; )

RecapWe only need to know two sets of equations

General form of Reparameterization

’a;i = a;i

’ab;ik = ab;ik

+ Mab;k

- Mab;k

+ Mba;i

- Mba;i

’b;k = b;k

Reparameterization of (a,b) in Belief Propagation

Mab;k = mini { a;i + ab;ik }

Mba;i = 0

Dynamic Programming

3 variables 2 variables + book-keeping

n variables (n-1) variables + book-keeping

Start from left, go to right

Reparameterize current edge (a,b)

Mab;k = mini { a;i + ab;ik }

’ab;ik = ab;ik+ Mab;k - Mab;k’b;k = b;k

Repeat

• LP Relaxation and its Dual

• TRW Message Passing

Outline

Integer Programming Formulation

min ∑a ∑i a;i ya;i + ∑(a,b) ∑ik ab;ik yab;ik

ya;i {0,1}

∑i ya;i = 1

yab;ik = ya;i yb;k

Integer Programming Formulation

min Ty

ya;i {0,1}

∑i ya;i = 1

yab;ik = ya;i yb;k

= [ … a;i …. ; … ab;ik ….]

y = [ … ya;i …. ; … yab;ik ….]

Linear Programming Relaxation

min Ty

ya;i {0,1}

∑i ya;i = 1

yab;ik = ya;i yb;k

Two reasons why we can’t solve this


min Ty

ya;i [0,1]

∑i ya;i = 1

yab;ik = ya;i yb;k

One reason why we can’t solve this


min Ty

ya;i [0,1]

∑i ya;i = 1

∑k yab;ik = ∑kya;i yb;k



min Ty

ya;i [0,1]

∑i ya;i = 1


= 1∑k yab;ik = ya;i∑k yb;k


min Ty

ya;i [0,1]

∑i ya;i = 1

∑k yab;ik = ya;i



min Ty

ya;i [0,1]

∑i ya;i = 1

∑k yab;ik = ya;i

No reason why we can’t solve this *

*memory requirements, time complexity

Dual

Let’s try to understand it intuitively

Dual of the LP RelaxationWainwright et al., 2001

Va Vb Vc

Vd Ve Vf

Vg Vh Vi

Va Vb Vc

Vd Ve Vf

Vg Vh Vi

Va Vb Vc

Vd Ve Vf

Vg Vh Vi

1

2

3

4 5 6

i


q*(1)

i

q*(2)

q*(3)

q*(4) q*(5) q*(6)

q*(i)

Dual of LP

Va Vb Vc

Vd Ve Vf

Vg Vh Vi

Va Vb Vc

Vd Ve Vf

Vg Vh Vi

Va Vb Vc

Vd Ve Vf

Vg Vh Vi

max


i

max q*(i)

I can easily compute q*(i)

I can easily maintain reparam constraint

So can I easily solve the dual?


• TRW Message Passing

Outline

Things to Remember

• Forward-pass computes min-marginals of root

• BP is exact for trees

• Every iteration provides a reparameterization

TRW Message PassingKolmogorov, 2006

Va Vb Vc

Vd Ve Vf

Vg Vh Vi

VaVb Vc

VdVe Vf

VgVh Vi

1

2

3

4 5 6

i

q*(i)

Pick a variable Va


i

q*(i)

Vc Vb Va

1c;0

1c;1

1b;0

1b;1

1a;0

1a;1

Va Vd Vg

4a;0

4a;1

4d;0

4d;1

4g;0

4g;1


1 + 4 + rest

q*(1) + q*(4) + K

Vc Vb Va Va Vd Vg

Reparameterize to obtain min-marginals of Va

1c;0

1c;1

1b;0

1b;1

1a;0

1a;1

4a;0

4a;1

4d;0

4d;1

4g;0

4g;1


’1 + ’4 + rest

Vc Vb Va

’1c;0

’1c;1

’1b;0

’1b;1

’1a;0

’1a;1

Va Vd Vg

’4a;0

’4a;1

’4d;0

’4d;1

’4g;0

’4g;1

One pass of Belief Propagation

q*(’1) + q*(’4) + K


’1 + ’4 + rest

Vc Vb Va Va Vd Vg

Remain the same

q*(’1) + q*(’4) + K

’1c;0

’1c;1

’1b;0

’1b;1

’1a;0

’1a;1

’4a;0

’4a;1

’4d;0

’4d;1

’4g;0

’4g;1


’1 + ’4 + rest

min{’1a;0,’1a;1} + min{’4a;0,’4a;1} + K

Vc Vb Va Va Vd Vg

’1c;0

’1c;1

’1b;0

’1b;1

’1a;0

’1a;1

’4a;0

’4a;1

’4d;0

’4d;1

’4g;0

’4g;1


’1 + ’4 + rest

Vc Vb Va Va Vd Vg

Compute average of min-marginals of Va

’1c;0

’1c;1

’1b;0

’1b;1

’1a;0

’1a;1

’4a;0

’4a;1

’4d;0

’4d;1

’4g;0

’4g;1

min{’1a;0,’1a;1} + min{’4a;0,’4a;1} + K


’1 + ’4 + rest

Vc Vb Va Va Vd Vg

’’a;0 = ’1a;0+ ’4a;0

2

’’a;1 = ’1a;1+ ’4a;1

2

’1c;0

’1c;1

’1b;0

’1b;1

’1a;0

’1a;1

’4a;0

’4a;1

’4d;0

’4d;1

’4g;0

’4g;1

min{’1a;0,’1a;1} + min{’4a;0,’4a;1} + K


’’1 + ’’4 + rest

Vc Vb Va Va Vd Vg

’1c;0

’1c;1

’1b;0

’1b;1

’’a;0

’’a;1

’’a;0

’’a;1

’4d;0

’4d;1

’4g;0

’4g;1

’’a;0 = ’1a;0+ ’4a;0

2

’’a;1 = ’1a;1+ ’4a;1

2

min{’1a;0,’1a;1} + min{’4a;0,’4a;1} + K


’’1 + ’’4 + rest

Vc Vb Va Va Vd Vg

’1c;0

’1c;1

’1b;0

’1b;1

’’a;0

’’a;1

’’a;0

’’a;1

’4d;0

’4d;1

’4g;0

’4g;1

’’a;0 = ’1a;0+ ’4a;0

2

’’a;1 = ’1a;1+ ’4a;1

2

min{’1a;0,’1a;1} + min{’4a;0,’4a;1} + K


Vc Vb Va Va Vd Vg

2 min{’’a;0, ’’a;1} + K

’1c;0

’1c;1

’1b;0

’1b;1

’’a;0

’’a;1

’’a;0

’’a;1

’4d;0

’4d;1

’4g;0

’4g;1

’’1 + ’’4 + rest

’’a;0 = ’1a;0+ ’4a;0

2

’’a;1 = ’1a;1+ ’4a;1

2


Vc Vb Va Va Vd Vg

’1c;0

’1c;1

’1b;0

’1b;1

’’a;0

’’a;1

’’a;0

’’a;1

’4d;0

’4d;1

’4g;0

’4g;1

min {p1+p2, q1+q2} min {p1, q1} + min {p2, q2}≥

2 min{’’a;0, ’’a;1} + K

’’1 + ’’4 + rest


Vc Vb Va Va Vd Vg

Objective function increases or remains constant

’1c;0

’1c;1

’1b;0

’1b;1

’’a;0

’’a;1

’’a;0

’’a;1

’4d;0

’4d;1

’4g;0

’4g;1

2 min{’’a;0, ’’a;1} + K

’’1 + ’’4 + rest

TRW Message Passing

Initialize i. Take care of reparam constraint

Choose random variable Va

Compute min-marginals of Va for all trees

Node-average the min-marginals

REPEAT

Kolmogorov, 2006

Can also do edge-averaging

• Preliminaries


• TRW Message Passing– Examples– Primal Solution– Results

Outline

Example 1

Va Vb

0

1 1

0

2

5

4

2l0

l1

Vb Vc

0

2 3

1

4

2

6

3

Vc Va

1

4 1

0

6

3

6

4

5

6

7

Pick variable Va. Reparameterize.

Example 1

Va Vb

-3

-2 -1

-2

5

7

4

2

Vb Vc

0

2 3

1

4

2

6

3

Vc Va

-3

1 -3

-3

6

3

10

7

5

6

7

Average the min-marginals of Va

l0

l1

Example 1

Va Vb

-3

-2 -1

-2

7.5

7

4

2

Vb Vc

0

2 3

1

4

2

6

3

Vc Va

-3

1 -3

-3

6

3

7.5

7

7

6

7

Pick variable Vb. Reparameterize.

l0

l1

Example 1

Va Vb

-7.5

-7 -5.5

-7

7.5

7

8.5

7

Vb Vc

-5

-3 -1

-3

9

6

6

3

Vc Va

-3

1 -3

-3

6

3

7.5

7

7

6

7

Average the min-marginals of Vb

l0

l1

Example 1

Va Vb

-7.5

-7 -5.5

-7

7.5

7

8.75

6.5

Vb Vc

-5

-3 -1

-3

8.75

6.5

6

3

Vc Va

-3

1 -3

-3

6

3

7.5

7

6.5

6.5

7 Value of dual does not increase

l0

l1

Example 1

Va Vb

-7.5

-7 -5.5

-7

7.5

7

8.75

6.5

Vb Vc

-5

-3 -1

-3

8.75

6.5

6

3

Vc Va

-3

1 -3

-3

6

3

7.5

7

6.5

6.5

7 Maybe it will increase for Vc

NO

l0

l1

Example 1

Va Vb

-7.5

-7 -5.5

-7

7.5

7

8.75

6.5

Vb Vc

-5

-3 -1

-3

8.75

6.5

6

3

Vc Va

-3

1 -3

-3

6

3

7.5

7

Strong Tree Agreement

Exact MAP Estimate

f1(a) = 0 f1(b) = 0 f2(b) = 0 f2(c) = 0 f3(c) = 0 f3(a) = 0

l0

l1

Example 2

Va Vb

0

1 1

0

2

5

2

2

Vb Vc

1

0 0

1

0

0

0

0

Vc Va

0

1 1

0

0

3

4

8

4

0

4

Pick variable Va. Reparameterize.

l0

l1

Example 2

Va Vb

-2

-1 -1

-2

4

7

2

2

Vb Vc

1

0 0

1

0

0

0

0

Vc Va

0

0 1

-1

0

3

4

9

4

0

4

Average the min-marginals of Va

l0

l1

Example 2

Va Vb

-2

-1 -1

-2

4

8

2

2

Vb Vc

1

0 0

1

0

0

0

0

Vc Va

0

0 1

-1

0

3

4

8

4

0

4 Value of dual does not increase

l0

l1

Example 2

Va Vb

-2

-1 -1

-2

4

8

2

2

Vb Vc

1

0 0

1

0

0

0

0

Vc Va

0

0 1

-1

0

3

4

8

4

0

4 Maybe it will decrease for Vb or Vc

NO

l0

l1

Example 2

Va Vb

-2

-1 -1

-2

4

8

2

2

Vb Vc

1

0 0

1

0

0

0

0

Vc Va

0

0 1

-1

0

3

4

8

f1(a) = 1 f1(b) = 1 f2(b) = 1 f2(c) = 0 f3(c) = 1 f3(a) = 1

f2(b) = 0 f2(c) = 1

Weak Tree Agreement

Not Exact MAP Estimate

l0

l1

Example 2

Va Vb

-2

-1 -1

-2

4

8

2

2

Vb Vc

1

0 0

1

0

0

0

0

Vc Va

0

0 1

-1

0

3

4

8

Weak Tree Agreement

Convergence point of TRW

l0

l1

f1(a) = 1 f1(b) = 1 f2(b) = 1 f2(c) = 0 f3(c) = 1 f3(a) = 1

f2(b) = 0 f2(c) = 1

• Preliminaries



Outline

Obtaining the Labelling

Only solves the dual. Primal solutions?

Va Vb Vc

Vd Ve Vf

Vg Vh Vi

’ = i

Fix the labelOf Va

Obtaining the Labelling

Only solves the dual. Primal solutions?

Va Vb Vc

Vd Ve Vf

Vg Vh Vi

’ = i

Fix the labelOf Vb

Continue in some fixed order

Meltzer et al., 2006

Computational Issues of TRW

• Speed-ups for some pairwise potentials

Basic Component is Belief Propagation

Felzenszwalb & Huttenlocher, 2004

• Memory requirements cut down by half

Kolmogorov, 2006

• Further speed-ups using monotonic chains

Kolmogorov, 2006

Theoretical Properties of TRW

• Always converges, unlike BP

Kolmogorov, 2006

• Strong tree agreement implies exact MAP

Wainwright et al., 2001

• Optimal MAP for two-label submodular problems

Kolmogorov and Wainwright, 2005

ab;00 + ab;11 ≤ ab;01 + ab;10

• Preliminaries



Outline

ResultsBinary Segmentation Szeliski et al. , 2008

Labels - {foreground, background}

Unary Potentials: -log(likelihood) using learnt fg/bg models

Pairwise Potentials: 0, if same labels

1 - exp(|da - db|), if different labels

ResultsBinary Segmentation



Szeliski et al. , 2008



TRW

ResultsBinary Segmentation



Szeliski et al. , 2008

Belief Propagation



ResultsStereo Correspondence Szeliski et al. , 2008

Labels - {disparities}

Unary Potentials: Similarity of pixel colours



ResultsSzeliski et al. , 2008





TRW

Stereo Correspondence

ResultsSzeliski et al. , 2008



Belief Propagation



Stereo Correspondence

ResultsNon-submodular problems Kolmogorov, 2006

BP TRW-S

30x30 grid K50

BP TRW-S

BP outperforms TRW-S

Code + Standard Data

http://vision.middlebury.edu/MRF

Documents

Discrete Optimization Lecture 3 – Part 1 M. Pawan Kumar [email protected] Slides available online