Introduction to Programming
Variable EliminationGraph-Based
PerspectiveProbabilisticGraphicalModelsInference
Daphne Koller1Initial GraphCDISGLJHDCDISGLJHD
Daphne KollerElimination as Graph Operation
Eliminate: C
CDISGLJHDInduced Markov network for the current set of
factorsDaphne KollerElimination as Graph OperationEliminate: D
CDISGLJHD
Induced Markov network for the current set of factorsDaphne
KollerElimination as Graph OperationEliminate: I
CDISGLJH
Induced Markov network for the current set of factorsDaphne
KollerElimination as Graph OperationEliminate: HCDISGLJH
Induced Markov network for the current set of factorsDaphne
KollerElimination as Graph OperationEliminate: GCDISGLJH
Induced Markov network for the current set of factorsDaphne
KollerElimination as Graph OperationEliminate: L,SCDISGLJH
Induced Markov network for the current set of factorsDaphne
KollerElimination as Graph OperationEliminate: L,SCDISGLJH
Induced Markov network for the current set of factorsDaphne
KollerInduced GraphThe induced graph I, over factors and ordering
:Undirected graphXi and Xj are connected if they appeared in the
same factor in a run of the VE algorithm using as the ordering
CDISGLJHDDaphne KollerCliques in the Induced GraphTheorem: Every
factor produced during VE is a clique in the induced graph
CDISGLJHD
Daphne Koller11Cliques in the Induced GraphTheorem: Every
(maximal) clique in the induced graph is a factor produced during
VE
CDISGLJHD
Daphne Koller12Cliques in the Induced GraphTheorem: Every
(maximal) clique in the induced graph is a factor produced during
VE
CDISGLJHDDaphne Koller13Induced WidthThe width of an induced
graph is the number of nodes in the largest clique in the graph
minus 1Minimal induced width of a graph K is
min(width(IK,))Provides a lower bound on best performance of VE to
a model factorizing over K
Daphne Koller14Finding Elimination OrderingsTheorem: For a graph
H, determining whether there exists an elimination ordering for H
with induced width K is NP-complete
Note: This NP-hardness result is distinct from the NP-hardness
result of inferenceEven given the optimal ordering, inference may
still be exponential
Daphne Koller15Finding Elimination OrderingsGreedy search using
heuristic cost functionAt each point, eliminate node with smallest
costPossible cost functions:min-neighbors: # neighbors in current
graphmin-weight: weight (# values) of factor formed min-fill:
number of new fill edgesweighted min-fill: total weight of new fill
edges (edge weight = product of weights of the 2 nodes)Daphne
Koller16Finding Elimination OrderingsTheorem: The induced graph is
triangulatedNo loops of length > 3 without a bridge
Can find elimination ordering by finding a low-width
triangulation of original graph HBDCADaphne Koller17Robot
Localization & Mapping
Square Root SAM, F. Dellaert and M. Kaess, IJRR, 2006Daphne
KollerRobot Localization & Mappingxzxz22xz...ttx110robot
posesensor observationL1L2L3xz33xz44Daphne Koller
Robot Localization & MappingSquare Root SAM, F. Dellaert and
M. Kaess, IJRR, 2006Daphne KollerThe associated factor graph, as
imported in mfqrEliminate Poses then Landmarks
Square Root SAM, F. Dellaert and M. Kaess, IJRR, 2006Daphne
KollerThe associated factor graph, as imported in mfqrEliminate
Landmarks then Poses
Square Root SAM, F. Dellaert and M. Kaess, IJRR, 2006Daphne
KollerThe associated factor graph, as imported in mfqrMin-Fill
EliminationSquare Root SAM, F. Dellaert and M. Kaess, IJRR,
2006
Daphne KollerSummaryVariable elimination can be viewed in terms
of transformations on undirected graphEliminating Z connects its
current neighborsSizes of cliques in resulting induced graph
directly correspond to algorithms complexityKeeping induced graph
simple provides useful heuristics for selecting elimination
orderingDaphne KollerEND END ENDDaphne Koller
