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Exact methods for submodular energies

Approximations for non-submodular energies

Move-making ( N_Variables >> N_Labels)

Inference for Learning

Linear Programming Relaxation

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 !!

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

Roadmap

Express MAP as an integer program

Relax to a linear program and solve

Round fractional solution to integers

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 ]

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

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

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

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

Integer Programming Formulation

Constraints

• Uniqueness Constraint

∑ xi = 1i Va

• Integer Constraints

xi {0,1}

X = x xT

Integer Programming Formulation

x* = argmin ∑ ui xi + ∑ Pij Xij

xi {0,1}

X = x xT

∑ xi = 1i Va

Roadmap

Express MAP as an integer program

Relax to a linear program and solve

Round fractional solution to integers

Integer Programming Formulation

x* = argmin ∑ ui xi + ∑ Pij Xij

∑ xi = 1i Va

xi {0,1}

X = x xT

Convex

Non-Convex

Integer Programming Formulation

x* = argmin ∑ ui xi + ∑ Pij Xij

∑ xi = 1i Va

xi [0,1]

X = x xT

Convex

Non-Convex

Integer Programming Formulation

x* = argmin ∑ ui xi + ∑ Pij Xij

∑ xi = 1i Va

xi [0,1]

Xij [0,1]

Convex

∑ Xij = xij Vb

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

Roadmap

Express MAP as an integer program

Relax to a linear program and solve

Round fractional solution to integers

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

Algorithms

Tree-reweighted message passing (TRW)

Max-product linear programming (MPLP)

Dual decomposition

Komodakis and Paragios, ICCV 2007