Uses of persistence for interpreting coarse instructions

ICRA Workshop on Emerging Topological Techniques in Robotics
20th May 2016
SubramanianRamamoorthy
SchoolofInforma-cs
TheUniversityofEdinburgh
Scenario:Human-RobotCo-Work
20/05/16
2
HowDoPeopleGiveInstruc-ons
forSpa-alNaviga-on?
HCRCMapTask:Instruc-onGiver’staskistocommunicatearoutetoaFollower,
whosemapmaydiffer.RouteisGiver’sgoalwhichtheFollowertriestoinfer.
20/05/16
A.Eshky,B.Allison,S.Ramamoorthy,M.Steedman,Agenera9vemodelforusersimula9on
inaspa9alnaviga9ondomain,InProc.EACL2014.
3
ExampleDataforMapTask
20/05/16
4
Genera-veModel:BehaviourinMapTask
[A.Eshkyetal.,EACL2014]
CoreQues9on:Canwehavemul9-scalestatees9ma9ontodealwith
evidencefrommul9plesuchmodali9es,ofvaryingcoarseness?
20/05/16
5
Representa-vePriorWork
•  Variableresolu-onpar-clefiltering[Verma,Thrun,Simmons
IJCAI‘03]:clumpstatestodefine‘macrostate’
•  Hierarchicalsubspacefiltering[e.g.,Brandaoetal.‘06]:track
subsetofvariablesseparatelyandaggregate
20/05/16
6
OurApproach
20/05/16
[M.Hawasly,F.T.Pokorny,S.Ramamoorthy,HierarchicalFilteringwithSpa9al
Abstrac9ons,Manuscriptinprepara-on]
7
HierarchicalAgglomera-veClustering
•  Iteratedopera-on:mergetwoclustersatlowerleveltogeta
singlenewcluster
•  Yieldsatreedatastructure
–  leavesaretheindividualdataitems
–  rootnodeistheclustermadebymergingalldatapoints
•  Orderofmergingdependsondistancebetweenclusters,such
thatthepairwiththesmallestdistanceismergedfirst.
•  Everynewclustercanbeassignedadistancevalueatwhichit
getscreated.
20/05/16
8
SimpleClusteringScheme
•  Collec-onofobjects(e.g.,trajectories): C = {c1 , c2 , ..., cM }
•  Distancematrix: D(i, j) = (ci , cj )
[
cu
•  Mergetocreatenewcluster: cij =
u2arg mini,j,i6=j Di,j
bij = Di,j
cij
•  Birthindex,,isdistancethresholdatwhichclass
startstoexist
ci (cj )
•  Deathindex,,isdistanceindexwhen
di = dj = Di,j
ceasestoexist
20/05/16
9
OutputofHierarchicalClustering
•  Treestructure,T hC, ⇢i
C : collection of all original/hierarchical classes
⇢ : C 7! C : maps class to its ‘parent’
⇢(ci ) = cj
•  If
then bj = di and ci ⇢ cj
ci 2 C
•  Treenodeisalive
when bi  b < di
Cb
denotethelevelby
20/05/16
10
FréchetDistance
•  Distancebetweentwocurves(orsurfaces)
•  ThediscreteFréchetdistanceisdefinedas
F (⌧1 , ⌧2 )
= inf max
↵,
j
E (⌧1 (↵(j)), ⌧2 (
(j)))
↵,
arere-parametrisa-onsthataligntrajectoriestoeach
otherpoint-wise
20/05/16
11
Illustra-veExample:HierarchicalClustering
withFréchetDistance
20/05/16
12
ClusteringbyTopology
20/05/16
13
Recap:Construc-ngFiltra-onsfromSamples
Observedtrajectoriesyield
samplesintheC-Space,
Then,considerthespace,
20/05/16
14
UnionsofBalls
20/05/16
15
UnionsofBalls
20/05/16
16
Delaunay–ČechComplex
IfwecantransformasimplicialcomplexKintoanothercomplexK0by
asequenceofelementarycollapses,thenthetwocomplexeshavethe
samehomotopytype.
AdiscreteMorsefunc-oncanencodesuchasimplicialcollapse,
whichisusedtoestablishtheaboveresult.
U.Bauer,H.Edelsbrunner.TheMorsetheoryofČechandDelaunayfiltra9ons.
InProc.Symp.Comp.Geometry,SOCG’14.
20/05/16
17
HomologyComputa-on
byMatrixReduc-on
a
a
d
e
a
i j
h
f b g c
b
c
d
b
c
g+h+i=[0000]T
h+i=[0110]T
d
e
f
g
h
i
1
1
0
0
0
0
1
1
0
1
0
0
1
1
0
1
0
0
1
1
j
e
0
f
0
g
1
h
1
i
1
j
[H.Edelsbrunner,J.Harer,Contemp.Math.453:257-282,2008.]
20/05/16
18
TrajectoryClassifica-oninSimplicialComplex
are not homotopy
equivalent in
20/05/16
19
PersistentHomologyGroups
Forafiltra9onofsimplicialcomplexes
,
definethep-thpersistenthomologygroups:
20/05/16
20
TwoTrajectoryClassesfor4-linkArm
Class“S”
20/05/16
Class“Z”
21
ClusteringEndEffectorTrajectories:
3-dim,97trajectories
Thereexistsafiltra-onintervalwith3classes
whichlatermergeintotwoclasses
20/05/16
22
TwoDiscoveredEquivalenceClasses
20/05/16
23
Recap:Approach
20/05/16
24
HierarchicalBeliefs-Intui-on
•  Probabilityofnodesinthetreeóprobabili-esassignedto
regionsinametricspace
•  Intui-on:ConsiderVoronoitessella-onof2-dimspaceby
pointsondiscretecurves
20/05/16
25
BankofPar-cleFilters(MPF)
1.  Samplenpar-cleswithequalweightsfromaprior,
(C0 ⇥ <d )
C0 = {c 2 C|bc = b0 }
2.  Buildtreeprobabili-es*
3.  Samplenewpar-clesforparentnodes:
Inaloop:
a.  Foreachpar-cleateachlevel,computenewposi-on:
b.  Forpar-clesatchosenlevel,updatewithobserva-ons*
c.  Propagateweightsalongtree&resample(atlowestlevel)*
20/05/16
26
Mo-onModels
l 
Amo-onmodelislearntforeveryclass(treenode)
✓i = P(z 0 |z, ci ), z, z 0 2 <d
l 
Thecollec-onoftrajectoriesdefine:
-  Eitheraglobalmodel,e.g.,usingGMM/GMR
-  Or,alocalisedmodel-theaveragevelocityofthe
trajectorypointsinaballaroundthestudiedpointz
20/05/16
27
Incorpora-ngEvidence,BaseCase
•  Atlevel0(i.e.,atthelevelofthefinestscale),compute
Euclideandistancebetweenobservedposi-onandclass
predic-on E (z, ⇠ t )
•  Updateweights,inverselypropor-onaltodistance
p(z t , c) : w /
20/05/16
log(
t
(z,
⇠
))
E
28
Incorpora-ngEvidence,
withCoarseObserva-ons
Firstfindallclassesthatarealiveatthelevelofobserva-on,
b⇠ + d ⇠
ˆ
C = {ci 2 C| bi 
< di }
2
Foreachoftheseclasses:
Computedistancebasedonbirthindexoffirstsharedparent
t
(c,
⇠
) = b̂
T
Foreverypar-cleinclassc,updateas
p(z t , c) : w /
log(
t
(c,
⇠
))
T
Rebuildtreeprobabili-esandnormalise
20/05/16
29
RebuildingProbabili-esintheTree
1.  Updatebasedoncomputedweights,w,&nodeprobability,
2.  Downwardpass:Recursivelyupdatechildrens’probabili-es
3.  Upwardpass:Recursivelyupdateparents’probabili-es
20/05/16
30
Recap:Approach
20/05/16
31
Experiment1:Qualita-veBehaviour
•  Wecharacterisethe
temporalbehaviourusinga
synthe-cdataset
•  Thedatamimicstypical
paternofmovementof
pedestriansinbuilt
environments
•  Weupdateallprobabili-es
inthetree
20/05/16
32
Results:BeliefsOverTimeandLevel
20/05/16
33
TimeEvolu-onofBeliefs
20/05/16
34
Experiment2:UnderstandingPerformance
MPFvs.BaselinePF
Averagenormaliseddistancefromtruetrajectory
•  Metric:distancebetweenmaximumaposterioriclassofthe
filterandthetruetrajectory,ascapturedbythetreedistance,
averagedover-me
•  Coarseobserva-onsareprovideduniformlyatrandom(50%
ofthe-metherewillalsobeacoarseobserva-on)
20/05/16
35
MPFvs.Baseline
20/05/16
36
Experiment2:UnderstandingPerformance
MPFvs.BaselinePF
Timetoconvergence
•  Metric:normalised-metaken(%)fores-matetobewithinε,
bysimilarityintreedistance,oftruetrajectory.
•  Bothversionsreceivefineobserva-onsforasmallburn-in
periodfollowedbyonlycoarseobserva-ons(whichbaseline
PFisunabletouse)
20/05/16
37
MPFvs.Baseline
20/05/16
38
OngoingExperimentwithUberDataset
20/05/16
39
Recap:Filtra-onsArisingfromSublevelSets
WithgeneralmodelssuchasGaussianProcesses,doesthisapproachoffer
leveragebyextrac-ngqualita-vestructuretomakelearningmoretractable?
20/05/16
40
ExampleSurfacesofInterest:
MDPAc-vityModels
[F.Previtalietal.,
ICRA2015,IROS16subm]
InverseRLmethods
canbeusedto
es-matecostfunc-ons
Realisedpathsareactuallydeterminesbyavaluefunc-on,expensivecomputa-on:
Cantopologicalcategorisa-onofrewardfunc-onsyieldfastercomputa-onofV?
04/06/16
41
Conclusions
•  Persistenceasdefinedin
TDAprovidesamul--scale
representa-onthatisripe
forcombina-onswith
probabili-es
•  Someuses:
–  Var.noisesensors/
actuators
–  Coarseinstruc-ons
20/05/16
42
Acknowledgements
ThisworkwassupportedbytheEUprojects
TOMSY(IST-FP7-270436)andTOPOSYS(ICT-FP7-18493)
Wehavebenefitedfromtheuseofequipmentwithinthe
EPSRCRobotariumResearchFacilityattheEdinburghCentreforRobo-cs.
20/05/16
43