M Pawan Kumar École Centrale Paris Joint work with Phil Torr Daphne Koller Metric Labeling Variables V V 1 V 2 V n Metric Labeling Variables V V ID: 783354
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Slide1
Multiplicative Boundsfor Metric Labeling
M. Pawan KumarÉcole Centrale Paris
Joint work with Phil
Torr
, Daphne Koller
Slide2Metric Labeling
Variables
V
= { V
1
, V
2
, …,
V
n
}
Slide3Metric Labeling
Variables V = { V1, V2, …, Vn}
Slide4Metric Labeling
VaVb
Labels
L
= {
l
1
,
l
2
, …,
l
h
}
Variables
V
= { V
1
, V
2
, …,
V
n
}
Labeling f: { 1, 2, …, n}
{1, 2, …, h}
E(f)
=
Σ
a
θa(f(a))
+ Σ(a,b) wabd
(f(a),f(b))
minf
θ
a(f(a))
θb(f(b))
w
abd(f(a),f(b))
w
ab
≥ 0
d is metric
Slide5Metric Labeling
VaVb
E(f)
min
f
NP hard
=
Σ
a
θ
a
(f(a))
+
Σ
(
a,b
)
w
ab
d
(
f(a),f(b))
Low-level vision applications
Slide6Minka. Expectation Propagation
for Approximate Bayesian Inference, UAI, 2001Murphy et al. Loopy Belief Propagation: An Empirical Study, UAI, 1999Winn et al. Variational Message Passing, JMLR, 2005Yedidia et al. Generalized Belief Propagation, NIPS, 2001Besag. On the Statistical Analysis of Dirty Pictures, JRSS, 1986Boykov et al. Fast Approximate Energy Minimization via Graph Cuts, PAMI, 2001Komodakis et al. Fast, Approximately Optimal Solutions for Single and Dynamic MRFs, CVPR, 2007Lempitsky et al. Fusion Moves for Markov Random Field Optimization, PAMI, 2010Chekuri et al. Approximation Algorithms for Metric Labeling, SODA, 2001Goemans et al. Improved Approximate Algorithms for Maximum-Cut, JACM, 1995Muramatsu et al. A New SOCP Relaxation for Max-Cut, JORJ, 2003Ravikumar
et al. QP Relaxations for
Metric Labeling
, ICML,
2006
Alahari
et al. Dynamic Hybrid
Algorithms for MAP Inference, PAMI 2010
Kohli et al. On Partial Optimality in Multilabel MRFs, ICML, 2008
Rother et al. Optimizing Binary MRFs via Extended Roof Duality, CVPR, 2007
...Approximate Algorithms
Slide7Outline
Linear Programming RelaxationMove-Making AlgorithmsComparisonRounding-based Moves
Slide8Integer Linear Program
Number of facets grows exponentially in problem size
Minimize a linear function over a set of feasible solutions
Indicator
x
a
(
i
)
{0,1} for each variable
V
a
and label li
Slide9Linear Programming Relaxation
Schlesinger, 1976; Chekuri et al., 2001; Wainwright et al., 2003Indicator xa(i) {0,1} for each variable Va and label li
Slide10Linear Programming Relaxation
Schlesinger, 1976; Chekuri et al., 2001; Wainwright et al., 2003Indicator xa(i)
[0,1] for each variable
V
a
and label l
i
Slide11Outline
Linear Programming RelaxationMove-Making AlgorithmsComparisonRounding-based Moves
Slide12Move-Making Algorithms
Space of All Labelingsf
Slide13Expansion Algorithm
Variables take label lα or retain current labelSlide courtesy Pushmeet KohliBoykov
,
Veksler
and
Zabih
, 2001
Slide14Expansion Algorithm
Sky
House
Tree
Ground
Initialize with Tree
Status:
Expand Ground
Expand House
Expand Sky
Slide courtesy
Pushmeet
Kohli
Variables take label
l
α
or
retain current label
Boykov
,
Veksler
and
Zabih
, 2001
Slide15Outline
Linear Programming RelaxationMove-Making AlgorithmsComparisonRounding-based Moves
Slide16Multiplicative Bounds
f*: Optimal Labelingf: Estimated LabelingΣa θa(f(a)) + Σ(a,b) sabd(f(a),f(b))
Σ
a
θ
a
(f*(a))
+
Σ(a,b)
sabd(f*(a),f*(b))
≥
Slide17Multiplicative Bounds
f*: Optimal Labelingf: Estimated Labeling≤BΣa
θ
a
(f(a))
+
Σ
(
a,b
) sab
d(f(a),f(b))
Σa θa(f*(a))
+ Σ(a,b) sabd(f*(a),
f*(b))
Slide18Multiplicative Bounds
Expansion
LP
Potts
2
2
Metric
2M
O(log h)
Truncated
Linear
2M
2 + √2
Truncated
Quadratic
2M
O(√M)
M = ratio of maximum and minimum distance
Slide19Outline
Linear Programming RelaxationMove-Making AlgorithmsComparisonRounding-based MovesComplete RoundingInterval RoundingHierarchical Rounding
Slide20Complete Rounding
Treat xa(i) [0,1] as probability that f(a) = i
Cumulative probability
y
a
(
i
)
=
Σ
j≤i
x
a(j)
0
y
a
(1)
y
a
(2)
y
a
(h) = 1
r
Generate a random number r
(0,1]
Assign the label next to r
y
a
(k)
y
a
(
i
)
Slide21Complete Move
Va
V
b
θ
ab
(
i,k
) =
s
ab
d
(
i,k
)
NP-hard
Slide22Complete Move
Va
V
b
θ
ab
(
i,k
) =
s
ab
d
’(
i,k
)
d’(
i,k
)
≥ d(
i,k
)
d’ is
submodular
Slide23Complete Move
Va
V
b
θ
ab
(
i,k
) =
s
ab
d
’(
i,k
)
d’(
i,k
)
≥ d(
i,k
)
d’ is
submodular
Slide24Complete Move
New problem can be solved using minimum cut
Same multiplicative bound as complete rounding
Multiplicative bound is tight
Slide25Outline
Linear Programming RelaxationMove-Making AlgorithmsComparisonRounding-based MovesComplete RoundingInterval RoundingHierarchical Rounding
Slide26Interval Rounding
Treat xa(i) [0,1] as probability that f(a) = i
Cumulative probability
y
a
(
i
)
=
Σ
j≤i
x
a(j)
0
y
a
(1)
y
a
(2)
y
a
(h) = 1
y
a
(k)
y
a
(
i
)
Choose an interval of length h’
Slide27Interval Rounding
Treat xa(i) [0,1] as probability that f(a) = i
Cumulative probability
y
a
(
i
)
=
Σ
j≤i
x
a(j)
r
Generate a random number r
(0,1]
Assign the label next to r
if it is within the interval
y
a
(k)
y
a
(
i
)
Choose an interval of length
h’
REPEAT
Slide28Interval Move
Va
V
b
θ
ab
(
i,k
) =
s
ab
d
(
i,k
)
Choose an interval of length h’
Slide29Interval Move
Va
V
b
θ
ab
(
i,k
) =
s
ab
d
(
i,k
)
Choose an interval of length h’
Add the current labels
Slide30Interval Move
Va
V
b
θ
ab
(
i,k
) =
s
ab
d
’(
i,k
)
Choose an interval of length h’
Add the current labels
d’(
i,k
)
≥ d(
i,k
)
d’ is
submodular
Solve to update labels
Repeat until convergence
Slide31Interval Move
Each problem can be solved using minimum cut
Same multiplicative bound as interval rounding
Multiplicative bound is tight
Kumar and
Torr
, NIPS 2008
Slide32Outline
Linear Programming RelaxationMove-Making AlgorithmsComparisonRounding-based MovesComplete RoundingInterval RoundingHierarchical Rounding
Slide33Hierarchical Rounding
L1L2l1l2l3
l
4
l
5
l
6
l
7
l
8
l
9
L
3
Hierarchical clustering of labels (e.g. r-HST metrics)
Slide34Hierarchical Rounding
L1L2l1l2l3
l
4
l
5
l
6
l
7
l
8
l
9
L
3
Assign variables to labels L
1
, L
2
or L
3
Move down the hierarchy until the leaf level
Slide35Hierarchical Rounding
L1L2l1l2l3
l
4
l
5
l
6
l
7
l
8
l
9
L
3
Assign variables to labels l
1
, l
2
or l
3
Slide36Hierarchical Rounding
L1L2l1l2l
3
l
4
l
5
l
6
l
7
l
8
l
9
L
3
Assign variables to labels l
4
, l
5
or l
6
Slide37Hierarchical Rounding
L1L2l1l2l
3
l
4
l
5
l
6
l
7
l
8
l
9
L
3
Assign variables to labels l
7
, l
8
or l
9
Slide38Hierarchical Move
L1L2l1l2l3
l
4
l
5
l
6
l
7
l
8
l
9
L
3
Hierarchical clustering of labels (e.g. r-HST metrics)
Slide39Hierarchical Move
L1L2l1l2l3
l
4
l
5
l
6
l
7
l
8
l
9
L
3
Obtain labeling f
1
restricted to labels {l
1
,l
2
,l
3
}
Slide40Hierarchical Move
L1L2l1l2l
3
l
4
l
5
l
6
l
7
l
8
l
9
L
3
Obtain labeling f
2
restricted to labels {l
4
,l
5
,l
6
}
Slide41Hierarchical Move
L1L2l1l2l
3
l
4
l
5
l
6
l
7
l
8
l
9
L
3
Obtain labeling f
3
restricted to labels {l
7
,l
8
,l
9
}
Slide42Hierarchical Move
L1L2L3
V
a
V
b
f
1
(a)
f
2
(a)
f
3
(a)
Move up the hierarchy until we reach the root
f
1
(b)
f
2
(b)
f
3
(b)
Slide43Hierarchical Move
Each problem can be solved using minimum cut
Same multiplicative bound as hierarchical rounding
Multiplicative bound is tight
Kumar and Koller, UAI 2009
Slide44Conclusion
Move-Making
LP
Potts
2
2
Metric
O(log h)
O(log h)
Truncated
Linear
2 + √2
2 + √2
Truncated
Quadratic
O(√M)
O(√M)
M = ratio of maximum and minimum distance
Slide45Open Problems
Moves for general rounding schemesHigher-order energy functionsBetter comparison criterion
Slide46Questions?
http://www.centrale-ponts.fr/personnel/pawan