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Multiplicative Bounds for Metric Labeling. 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 1 , V 2 , …, V n }. Metric Labeling. w ab d ( f(a),f(b)). θ b (f(b)). - PowerPoint PPT Presentation

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Multiplicative Boundsfor Metric LabelingM. Pawan Kumar

cole Centrale ParisJoint work with Phil Torr, Daphne KollerMetric LabelingVariables V = { V1, V2, , Vn}Metric LabelingVariables V = { V1, V2, , Vn}Metric LabelingVaVbLabels L = { l1, l2, , lh}Variables V = { V1, V2, , Vn}Labeling f: { 1, 2, , n} {1, 2, , h}E(f)= a a(f(a))+ (a,b) wabd(f(a),f(b))minfa(f(a))b(f(b))wabd(f(a),f(b))wab 0d is metricMetric LabelingVaVbE(f)minfNP hard= a a(f(a))+ (a,b) wabd(f(a),f(b))Low-level vision applicationsMinka. 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, 2006Alahari et al. Dynamic Hybrid Algorithms for MAP Inference, PAMI 2010Kohli et al. On Partial Optimality in Multilabel MRFs, ICML, 2008Rother et al. Optimizing Binary MRFs via Extended Roof Duality, CVPR, 2007 ...Approximate AlgorithmsOutlineLinear Programming Relaxation

Move-Making Algorithms

Comparison

Rounding-based MovesInteger Linear ProgramNumber of facets grows exponentially in problem sizeMinimize a linear function over a set of feasible solutionsIndicator xa(i) {0,1} for each variable Va and label li Linear Programming RelaxationSchlesinger, 1976; Chekuri et al., 2001; Wainwright et al., 2003Indicator xa(i) {0,1} for each variable Va and label li Linear Programming RelaxationSchlesinger, 1976; Chekuri et al., 2001; Wainwright et al., 2003Indicator xa(i) [0,1] for each variable Va and label li OutlineLinear Programming Relaxation

Move-Making Algorithms

Comparison

Rounding-based MovesMove-Making AlgorithmsSpace of All LabelingsfExpansion AlgorithmVariables take label l or retain current labelSlide courtesy Pushmeet KohliBoykov, Veksler and Zabih, 2001Expansion Algorithm

SkyHouseTreeGroundInitialize with TreeStatus:Expand GroundExpand HouseExpand SkySlide courtesy Pushmeet KohliVariables take label l or retain current labelBoykov, Veksler and Zabih, 2001OutlineLinear Programming Relaxation

Move-Making Algorithms

Comparison

Rounding-based MovesMultiplicative Boundsf*: Optimal Labelingf: Estimated Labelinga a(f(a)) + (a,b) sabd(f(a),f(b))a a(f*(a)) + (a,b) sabd(f*(a),f*(b))Multiplicative Boundsf*: Optimal Labelingf: Estimated LabelingBa a(f(a)) + (a,b) sabd(f(a),f(b))a a(f*(a)) + (a,b) sabd(f*(a),f*(b))Multiplicative BoundsExpansionLPPotts22Metric2MO(log h)TruncatedLinear2M2 + 2TruncatedQuadratic2MO(M)M = ratio of maximum and minimum distanceOutlineLinear Programming Relaxation

Move-Making Algorithms

Comparison

Rounding-based MovesComplete RoundingInterval RoundingHierarchical RoundingComplete RoundingTreat xa(i) [0,1] as probability that f(a) = i Cumulative probability ya(i) = ji xa(j)0ya(1)ya(2)ya(h) = 1rGenerate a random number r (0,1]Assign the label next to rya(k)ya(i)Complete MoveVaVbab(i,k) = sabd(i,k)

NP-hardComplete MoveVaVbab(i,k) = sabd(i,k)d(i,k) d(i,k)d is submodular

Complete MoveVaVbab(i,k) = sabd(i,k)d(i,k) d(i,k)d is submodular

Complete MoveNew problem can be solved using minimum cutSame multiplicative bound as complete roundingMultiplicative bound is tightOutlineLinear Programming Relaxation

Move-Making Algorithms

Comparison

Rounding-based MovesComplete RoundingInterval RoundingHierarchical RoundingInterval RoundingTreat xa(i) [0,1] as probability that f(a) = i Cumulative probability ya(i) = ji xa(j)0ya(1)ya(2)ya(h) = 1ya(k)ya(i)Choose an interval of length hInterval RoundingTreat xa(i) [0,1] as probability that f(a) = i Cumulative probability ya(i) = ji xa(j)rGenerate a random number r (0,1]Assign the label next to r if it is within the intervalya(k)ya(i)Choose an interval of length hREPEATInterval MoveVaVbab(i,k) = sabd(i,k)Choose an interval of length hInterval MoveVaVbab(i,k) = sabd(i,k)Choose an interval of length hAdd the current labelsInterval MoveVaVbab(i,k) = sabd(i,k)Choose an interval of length hAdd the current labelsd(i,k) d(i,k)d is submodularSolve to update labelsRepeat until convergenceInterval MoveEach problem can be solved using minimum cutSame multiplicative bound as interval roundingMultiplicative bound is tightKumar and Torr, NIPS 2008OutlineLinear Programming Relaxation

Move-Making Algorithms

Comparison

Rounding-based MovesComplete RoundingInterval RoundingHierarchical RoundingHierarchical RoundingL1L2l1l2l3l4l5l6l7l8l9L3Hierarchical clustering of labels (e.g. r-HST metrics)Hierarchical RoundingL1L2l1l2l3l4l5l6l7l8l9L3Assign variables to labels L1, L2 or L3Move down the hierarchy until the leaf levelHierarchical RoundingL1L2l1l2l3l4l5l6l7l8l9L3Assign variables to labels l1, l2 or l3Hierarchical RoundingL1L2l1l2l3l4l5l6l7l8l9L3Assign variables to labels l4, l5 or l6Hierarchical RoundingL1L2l1l2l3l4l5l6l7l8l9L3Assign variables to labels l7, l8 or l9Hierarchical MoveL1L2l1l2l3l4l5l6l7l8l9L3Hierarchical clustering of labels (e.g. r-HST metrics)Hierarchical MoveL1L2l1l2l3l4l5l6l7l8l9L3Obtain labeling f1 restricted to labels {l1,l2,l3}Hierarchical MoveL1L2l1l2l3l4l5l6l7l8l9L3Obtain labeling f2 restricted to labels {l4,l5,l6}Hierarchical MoveL1L2l1l2l3l4l5l6l7l8l9L3Obtain labeling f3 restricted to labels {l7,l8,l9}Hierarchical MoveL1L2L3VaVbf1(a)f2(a)f3(a)Move up the hierarchy until we reach the rootf1(b)f2(b)f3(b)Hierarchical MoveEach problem can be solved using minimum cutSame multiplicative bound as hierarchical roundingMultiplicative bound is tightKumar and Koller, UAI 2009ConclusionMove-MakingLPPotts22MetricO(log h)O(log h)TruncatedLinear2 + 22 + 2TruncatedQuadraticO(M)O(M)M = ratio of maximum and minimum distanceOpen ProblemsMoves for general rounding schemes

Higher-order energy functions

Better comparison criterionQuestions?http://www.centrale-ponts.fr/personnel/pawan