So far ...
description
Transcript of So far ...
![Page 1: So far ...](https://reader036.fdocuments.net/reader036/viewer/2022062305/568162a7550346895dd32751/html5/thumbnails/1.jpg)
So far ...
Exact methods for submodular energies
Approximations for non-submodular energies
Move-making ( N_Variables >> N_Labels)
![Page 2: So far ...](https://reader036.fdocuments.net/reader036/viewer/2022062305/568162a7550346895dd32751/html5/thumbnails/2.jpg)
![Page 3: So far ...](https://reader036.fdocuments.net/reader036/viewer/2022062305/568162a7550346895dd32751/html5/thumbnails/3.jpg)
Inference for Learning
Linear Programming Relaxation
![Page 4: So far ...](https://reader036.fdocuments.net/reader036/viewer/2022062305/568162a7550346895dd32751/html5/thumbnails/4.jpg)
Linear Integer Programming
minx g0Tx
s.t. giTx ≤ 0
hiTx = 0
Linear function
Linear constraints
Linear constraints
x is a vector of integers
For example, x {0,1}N
Hard to solve !!
![Page 5: So far ...](https://reader036.fdocuments.net/reader036/viewer/2022062305/568162a7550346895dd32751/html5/thumbnails/5.jpg)
Linear Programming
minx g0Tx
s.t. giTx ≤ 0
hiTx = 0
Linear function
Linear constraints
Linear constraints
x is a vector of reals
Easy to solve!!
For example, x [0,1]N
Relaxation
![Page 6: So far ...](https://reader036.fdocuments.net/reader036/viewer/2022062305/568162a7550346895dd32751/html5/thumbnails/6.jpg)
Roadmap
Express MAP as an integer program
Relax to a linear program and solve
Round fractional solution to integers
![Page 7: So far ...](https://reader036.fdocuments.net/reader036/viewer/2022062305/568162a7550346895dd32751/html5/thumbnails/7.jpg)
2
5
4
2
0
1 3
0V1 V2
Label ‘0’
Label ‘1’Unary Cost
Integer Programming Formulation
Unary Cost Vector u = [ 5
Cost of V1 = 0
2
Cost of V1 = 1
; 2 4 ]
![Page 8: So far ...](https://reader036.fdocuments.net/reader036/viewer/2022062305/568162a7550346895dd32751/html5/thumbnails/8.jpg)
2
5
4
2
0
1 3
0V1 V2
Label ‘0’
Label ‘1’Unary Cost
Unary Cost Vector u = [ 5 2 ; 2 4 ]T
Label vector x = [ 0
V1 0
1
V1 = 1
; 1 0 ]T
Integer Programming Formulation
![Page 9: So far ...](https://reader036.fdocuments.net/reader036/viewer/2022062305/568162a7550346895dd32751/html5/thumbnails/9.jpg)
2
5
4
2
0
1 3
0V1 V2
Label ‘0’
Label ‘1’Unary Cost
Unary Cost Vector u = [ 5 2 ; 2 4 ]T
Label vector x = [ 0 1 ; 1 0 ]T
Sum of Unary Costs = ∑i ui xi
Integer Programming Formulation
![Page 10: So far ...](https://reader036.fdocuments.net/reader036/viewer/2022062305/568162a7550346895dd32751/html5/thumbnails/10.jpg)
2
5
4
2
0
1 3
0V1 V2
Label ‘0’
Label ‘1’Pairwise Cost
Integer Programming Formulation
0 Cost of V1 = 0 and V1 = 00
00
0Cost of V1 = 0 and V2 = 0
3
Cost of V1 = 0 and V2 = 11 0
000 0
103 0
Pairwise Cost Matrix P
![Page 11: So far ...](https://reader036.fdocuments.net/reader036/viewer/2022062305/568162a7550346895dd32751/html5/thumbnails/11.jpg)
2
5
4
2
0
1 3
0V1 V2
Label ‘0’
Label ‘1’Pairwise Cost
Integer Programming Formulation
Pairwise Cost Matrix P
0 0
00
0 3
1 000
0 010
3 0
Sum of Pairwise Costs∑i<j Pij xixj
= ∑i<j Pij Xij
X = xxT
![Page 12: So far ...](https://reader036.fdocuments.net/reader036/viewer/2022062305/568162a7550346895dd32751/html5/thumbnails/12.jpg)
Integer Programming Formulation
Constraints
• Uniqueness Constraint
∑ xi = 1i Va
• Integer Constraints
xi {0,1}
X = x xT
![Page 13: So far ...](https://reader036.fdocuments.net/reader036/viewer/2022062305/568162a7550346895dd32751/html5/thumbnails/13.jpg)
Integer Programming Formulation
x* = argmin ∑ ui xi + ∑ Pij Xij
xi {0,1}
X = x xT
∑ xi = 1i Va
![Page 14: So far ...](https://reader036.fdocuments.net/reader036/viewer/2022062305/568162a7550346895dd32751/html5/thumbnails/14.jpg)
Roadmap
Express MAP as an integer program
Relax to a linear program and solve
Round fractional solution to integers
![Page 15: So far ...](https://reader036.fdocuments.net/reader036/viewer/2022062305/568162a7550346895dd32751/html5/thumbnails/15.jpg)
Integer Programming Formulation
x* = argmin ∑ ui xi + ∑ Pij Xij
∑ xi = 1i Va
xi {0,1}
X = x xT
Convex
Non-Convex
![Page 16: So far ...](https://reader036.fdocuments.net/reader036/viewer/2022062305/568162a7550346895dd32751/html5/thumbnails/16.jpg)
Integer Programming Formulation
x* = argmin ∑ ui xi + ∑ Pij Xij
∑ xi = 1i Va
xi [0,1]
X = x xT
Convex
Non-Convex
![Page 17: So far ...](https://reader036.fdocuments.net/reader036/viewer/2022062305/568162a7550346895dd32751/html5/thumbnails/17.jpg)
Integer Programming Formulation
x* = argmin ∑ ui xi + ∑ Pij Xij
∑ xi = 1i Va
xi [0,1]
Xij [0,1]
Convex
∑ Xij = xij Vb
![Page 18: So far ...](https://reader036.fdocuments.net/reader036/viewer/2022062305/568162a7550346895dd32751/html5/thumbnails/18.jpg)
Linear Programming Formulation
x* = argmin ∑ ui xi + ∑ Pij Xij
∑ xi = 1i Va
xi [0,1]
Xij [0,1]
Convex
∑ Xij = xij Vb
Schlesinger, 76; Chekuri et al., 01; Wainwright et al. , 01
![Page 19: So far ...](https://reader036.fdocuments.net/reader036/viewer/2022062305/568162a7550346895dd32751/html5/thumbnails/19.jpg)
Roadmap
Express MAP as an integer program
Relax to a linear program and solve
Round fractional solution to integers
![Page 20: So far ...](https://reader036.fdocuments.net/reader036/viewer/2022062305/568162a7550346895dd32751/html5/thumbnails/20.jpg)
Properties
Dominate many convex relaxations
Best known multiplicative bounds
2 for Potts (uniform) energies
2 + √2 for Truncated linear energies
O(log n) for metric labelingMatched by move-making
Kumar and Torr, 2008; Kumar and Koller, UAI 2009
Kumar, Kolmogorov and Torr, 2007
![Page 21: So far ...](https://reader036.fdocuments.net/reader036/viewer/2022062305/568162a7550346895dd32751/html5/thumbnails/21.jpg)
Algorithms
Tree-reweighted message passing (TRW)
Max-product linear programming (MPLP)
Dual decomposition
Komodakis and Paragios, ICCV 2007