Multiplicative Bounds for Metric Labeling
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Transcript of Multiplicative Bounds for Metric Labeling
Multiplicative Boundsfor Metric Labeling
M. Pawan Kumar
École Centrale Paris
Joint work with Phil Torr, Daphne Koller
Metric Labeling
Variables V = { V1, V2, …, Vn}
Metric Labeling
Variables V = { V1, V2, …, Vn}
Metric Labeling
Va Vb
Labels L = { l1, l2, …, lh}Variables V = { V1, V2, …, Vn}
Labeling f: { 1, 2, …, n} {1, 2, …, h}
E(f) = Σa θa(f(a)) + Σ(a,b) wabd(f(a),f(b))minf
θa(f(a))
θb(f(b))wabd(f(a),f(b))
wab ≥ 0
d is metric
Metric Labeling
Va Vb
E(f)minf
NP hard
= Σa θa(f(a)) + Σ(a,b) wabd(f(a),f(b))
Low-level vision applications
Minka. Expectation Propagation for Approximate Bayesian Inference, UAI, 2001 Murphy et al. Loopy Belief Propagation: An Empirical Study, UAI, 1999 Winn et al. Variational Message Passing, JMLR, 2005 Yedidia et al. Generalized Belief Propagation, NIPS, 2001 Besag. On the Statistical Analysis of Dirty Pictures, JRSS, 1986 Boykov et al. Fast Approximate Energy Minimization via Graph Cuts, PAMI, 2001 Komodakis et al. Fast, Approximately Optimal Solutions for Single and Dynamic MRFs, CVPR, 2007 Lempitsky et al. Fusion Moves for Markov Random Field Optimization, PAMI, 2010 Chekuri et al. Approximation Algorithms for Metric Labeling, SODA, 2001 Goemans et al. Improved Approximate Algorithms for Maximum-Cut, JACM, 1995 Muramatsu et al. A New SOCP Relaxation for Max-Cut, JORJ, 2003 Ravikumar 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
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Approximate Algorithms
Outline
• Linear Programming Relaxation
• Move-Making Algorithms
• Comparison
• Rounding-based Moves
Integer Linear Program
Number of facets grows exponentially in problem size
Minimize a linear function over a set of feasible solutions
Indicator xa(i) {0,1} for each variable Va and label li
Linear Programming Relaxation
Schlesinger, 1976; Chekuri et al., 2001; Wainwright et al., 2003
Indicator xa(i) {0,1} for each variable Va and label li
Linear Programming Relaxation
Schlesinger, 1976; Chekuri et al., 2001; Wainwright et al., 2003
Indicator xa(i) [0,1] for each variable Va and label li
Outline
• Linear Programming Relaxation
• Move-Making Algorithms
• Comparison
• Rounding-based Moves
Move-Making AlgorithmsSpace of All Labelings
f
Expansion AlgorithmVariables take label lα or retain current label
Slide courtesy Pushmeet Kohli Boykov, Veksler and Zabih, 2001
Expansion Algorithm
SkyHouse
TreeGround
Initialize with TreeStatus: Expand GroundExpand HouseExpand Sky
Slide courtesy Pushmeet Kohli
Variables take label lα or retain current label
Boykov, Veksler and Zabih, 2001
Outline
• Linear Programming Relaxation
• Move-Making Algorithms
• Comparison
• Rounding-based Moves
Multiplicative Boundsf*: Optimal Labeling f: Estimated Labeling
Σa θa(f(a)) + Σ(a,b) sabd(f(a),f(b))
Σa θa(f*(a)) + Σ(a,b) sabd(f*(a),f*(b))
≥
Multiplicative Boundsf*: Optimal Labeling f: Estimated Labeling
≤B
Σa θa(f(a)) + Σ(a,b) sabd(f(a),f(b))
Σa θa(f*(a)) + Σ(a,b) sabd(f*(a),f*(b))
Multiplicative Bounds
Expansion LP
Potts 2 2
Metric 2M O(log h)
TruncatedLinear
2M 2 + √2
TruncatedQuadratic
2M O(√M)
M = ratio of maximum and minimum distance
Outline
• Linear Programming Relaxation
• Move-Making Algorithms
• Comparison
• Rounding-based Moves– Complete Rounding– Interval Rounding– Hierarchical Rounding
Complete Rounding
Treat xa(i) [0,1] as probability that f(a) = i
Cumulative probability ya(i) = Σj≤i xa(j)
0 ya(1) ya(2) ya(h) = 1
r
Generate a random number r (0,1]
Assign the label next to r
ya(k)ya(i)
Complete Move
Va Vb
θab(i,k) = sabd(i,k) NP-hard
Complete Move
Va Vb
θab(i,k) = sabd’(i,k)
d’(i,k) ≥ d(i,k)
d’ is submodular
Complete Move
Va Vb
θab(i,k) = sabd’(i,k)
d’(i,k) ≥ d(i,k)
d’ is submodular
Complete Move
New problem can be solved using minimum cut
Same multiplicative bound as complete rounding
Multiplicative bound is tight
Outline
• Linear Programming Relaxation
• Move-Making Algorithms
• Comparison
• Rounding-based Moves– Complete Rounding– Interval Rounding– Hierarchical Rounding
Interval Rounding
Treat xa(i) [0,1] as probability that f(a) = i
Cumulative probability ya(i) = Σj≤i xa(j)
0 ya(1) ya(2) ya(h) = 1ya(k)ya(i)
Choose an interval of length h’
Interval Rounding
Treat xa(i) [0,1] as probability that f(a) = i
Cumulative probability ya(i) = Σj≤i xa(j)
r
Generate a random number r (0,1]
Assign the label next to r if it is within the interval
ya(k)ya(i)
Choose an interval of length h’ REPEAT
Interval Move
Va Vb
θab(i,k) = sabd(i,k)
Choose an interval of length h’
Interval Move
Va Vb
θab(i,k) = sabd(i,k)
Choose an interval of length h’
Add the current labels
Interval Move
Va Vb
θab(i,k) = sabd’(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
Interval Move
Each problem can be solved using minimum cut
Same multiplicative bound as interval rounding
Multiplicative bound is tight
Kumar and Torr, NIPS 2008
Outline
• Linear Programming Relaxation
• Move-Making Algorithms
• Comparison
• Rounding-based Moves– Complete Rounding– Interval Rounding– Hierarchical Rounding
Hierarchical Rounding
L1 L2
l1 l2 l3 l4 l5 l6 l7 l8 l9
L3
Hierarchical clustering of labels (e.g. r-HST metrics)
Hierarchical Rounding
L1 L2
l1 l2 l3 l4 l5 l6 l7 l8 l9
L3
Assign variables to labels L1, L2 or L3
Move down the hierarchy until the leaf level
Hierarchical Rounding
L1 L2
l1 l2 l3 l4 l5 l6 l7 l8 l9
L3
Assign variables to labels l1, l2 or l3
Hierarchical Rounding
L1 L2
l1 l2 l3 l4 l5 l6 l7 l8 l9
L3
Assign variables to labels l4, l5 or l6
Hierarchical Rounding
L1 L2
l1 l2 l3 l4 l5 l6 l7 l8 l9
L3
Assign variables to labels l7, l8 or l9
Hierarchical Move
L1 L2
l1 l2 l3 l4 l5 l6 l7 l8 l9
L3
Hierarchical clustering of labels (e.g. r-HST metrics)
Hierarchical Move
L1 L2
l1 l2 l3 l4 l5 l6 l7 l8 l9
L3
Obtain labeling f1 restricted to labels {l1,l2,l3}
Hierarchical Move
L1 L2
l1 l2 l3 l4 l5 l6 l7 l8 l9
L3
Obtain labeling f2 restricted to labels {l4,l5,l6}
Hierarchical Move
L1 L2
l1 l2 l3 l4 l5 l6 l7 l8 l9
L3
Obtain labeling f3 restricted to labels {l7,l8,l9}
Hierarchical Move
L1 L2 L3
Va Vb
f1(a)
f2(a)
f3(a)
Move up the hierarchy until we reach the root
f1(b)
f2(b)
f3(b)
Hierarchical Move
Each problem can be solved using minimum cut
Same multiplicative bound as hierarchical rounding
Multiplicative bound is tight
Kumar and Koller, UAI 2009
ConclusionMove-Making LP
Potts 2 2
Metric O(log h) O(log h)
TruncatedLinear
2 + √2 2 + √2
TruncatedQuadratic
O(√M) O(√M)
M = ratio of maximum and minimum distance
Open Problems
• Moves for general rounding schemes
• Higher-order energy functions
• Better comparison criterion
Questions?
http://www.centrale-ponts.fr/personnel/pawan