Riemannian Metric Learning for Symmetric Positive Definite Matrices

0.09 MB·English

by  Raviteja Vemulapalli

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Riemannian Metric Learning for Symmetric Positive
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Definite Matrices

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RavitejaVemulapalli,DavidW.Jacobs
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UniversityofMaryland,CollegePark

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Abstract. Overthepastfewyears,symmetricpositivedefinitematrices(SPD)
.
s have been receiving considerable attention from computer vision community.
c Though various distance measures have been proposed inthe past for compar-
[
ing SPD matrices, the two most widely-used measures areaffine-invariant dis-

1 tance and log-Euclidean distance. This is because these two measures are true
v geodesicdistancesinducedbyRiemanniangeometry.Inthiswork,wefocuson
3 thelog-EuclideanRiemanniangeometryandproposeadata-drivenapproachfor
9 learningRiemannianmetrics/geodesicdistancesforSPDmatrices.Weshowthat
3 thegeodesicdistancelearnedusingtheproposedapproachperformsbetterthan
2 variousexistingdistancemeasureswhenevaluatedonfacematchingandcluster-
0
ingtasks.
.
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5Notations
1
:
v – I denotestheidentitymatrixofappropriatesize.
i
X
– h, idenotesaninnerproduct.
r
a – S denotesthesetofn×nsymmetricmatrices.
n
– S++denotesthesetofn×nsymmetricpositivedefinitematrices.
n
– T MdenotesthetangentspacetothemanifoldMatthepointp∈M.
p
– kk denotesthematrixFrobeniusnorm.
F
– Chol(P)denotesthelowertriangularmatrixobtainedfromtheCholeskydecompo-
sitionofamatrixP.
– exp()andlog()denotematrixexponentialandlogarithmrespectively.
– ∂ and ∂2 representpartialderivatives.
∂x ∂x2
2 RavitejaVemulapalli,DavidW.Jacobs
1 Introduction
Manycomputervisionapplicationsinvolvefeaturesthatobeyspecificconstraints.Such
featuresoftenlieinnon-Euclideanspaces,wheretheunderlyingdistancemetricisnot
theregularℓ norm.Forinstance,popularfeatureslikeshapes,rotationmatrices,linear
2
subspaces, symmetric positive definite (SPD) matrices, etc. are known to lie on Rie-
mannianmanifolds.Insuchcases,oneneedstodevelopinferencetechniquesthatmake
useoftheunderlyingmanifoldstructure.
Overthepastfewyears,manifoldshavebeenreceivingconsiderableattentionfrom
thecomputervisioncommunity.Inthiswork,wefocusourattentiononthesetofSPD
matrices.ExamplesofSPD matricesin computervisionincludediffusiontensors[1],
structuretensors[2]andcovarianceregiondescriptors[3].Diffusiontensorsarisenat-
urally in medical imaging [1]. In diffusion tensor magnetic resonance imaging (DT-
MRI),waterdiffusionintissuesisrepresentedbyadiffusiontensorcharacterizingthe
anisotropywithinthetissue.Inopticalflowestimationandmotionsegmentation,struc-
turetensorsareoftenemployedtoencodeimportantimagefeatures,suchastextureand
motion[2].Covarianceregiondescriptorsare usedin textureclassification [3],object
detection[4],objecttracking,actionrecognitionandfacerecognition[5].Therearesev-
eraladvantagesofusingcovariancematricesasregiondescriptors.Covariancematrices
provideanaturalwayoffusingmultiplefeatureswhichmightbecorrelated.Thediag-
onalentriesofacovariancematrixrepresentthevarianceofindividualfeaturesandthe
non-diagonalentries represent the cross correlations. The noise corrupting individual
samplesislargelyfilteredoutwithanaveragingfilter duringcovariancecomputation.
Covariancematricesarelowdimensionalcomparedtojointfeaturehistograms.Covari-
ance matrices do not have any informationregardingthe orderingand the numberof
points.Thisimpliesacertainlevelofscaleandrotationinvarianceovertheregionsin
differentimages.
Variousdistancemeasureshavebeenproposedintheliteratureforthecomparison
of SPD matrices. Among them, the two most widely-used distance measures are the
affine-invariantdistance[1]andthelog-Frobeniusdistance[6](alsoreferredtoaslog-
Euclidean distance in the literature).The main reason for their popularityis that they
aregeodesicdistancesinducedbyRiemannianmetrics.
Thelog-Euclideanframework[6]proposedbyArsignyet.al.definesaclassofRie-
mannianmetrics,ratherthanasinglemetric,calledlog-EuclideanRiemannianmetrics.
Accordingtothisframework,anyinnerproducth, idefinedonT S++ ={log(P)|P ∈
I n
S++}=S extendedtoS++byleft-orright-multiplicationisabi-invariantRieman-
n n n
nianmetric.Equippedwiththisbi-invariantmetric,thespaceofSPDmatricesisaflat
Riemannianspaceandthegeodesicdistancecorrespondingtothisbi-invariantRieman-
nianmetricisequaltothedistanceinducedbyh, iinT S++.Surprisingly,thisremark-
I n
ableresulthasnotbeenusedbythecomputervisioncommunity.SinceT S++ = S
I n n
is avectorspace,thisresultallowsusto learnlog-EuclideanRiemannianmetricsand
corresponding log-Euclideangeodesic distances from the data by using Mahalanobis
distancelearningtechniqueslikeinformation-theoreticmetriclearning(ITML)[7]and
large margin nearest neighbor distance learning [8] in T S++. In this work, we ex-
I n
plorethisideaofdatadrivenRiemannianmetrics/geodesicdistancesforthesetofSPD
matrices.ForlearningMahalanobisdistancesinT S++ weusetheITMLtechnique.
I n
RiemannianMetricLearningforSPDMatrices 3
Organization:Insection2,weprovidea briefoverviewofvariousdistancemea-
sures used in the literature to compare SPD matrices. We briefly explain the ITML
techniqueinsection3andpresentourapproachforlearninglog-EuclideanRiemannian
metrics/log-Euclideangeodesicdistancesfromthedatainsection4.Weprovidesome
experimentalresultsinsection5andconcludethepaperinsection6.
2 Distances to compare SPD matrices
Variousdistance measures have been used in the literature to compareSPD matrices.
Each distance has been derived from differentgeometrical, statistical or information-
theoreticconsiderations.Thoughmanyofthesedistancestrytocapturethenon-linearity
ofSPDmatrices,notallofthemaregeodesicdistancesinducedbyRiemannianmetrics.
Tables1 and 2summarizethese distancesandtheirproperties.Amongthem,the log-
Frobeniusdistance[6]andtheaffine-invariantdistance[1]arethemostpopularones.
3 Mahalanobisdistance learning using ITML
Informationtheoreticmetriclearning[7]is a techniqueforlearningMahalanobisdis-
tance functions from the data based on similarity and dissimilarity constraints. Let
{x }N beasetofN pointsinRd.GivenpairsofsimilarpointsSandpairsofdissim-
i i=1
ilarpointsD,theaimofITMListolearnanSPDmatrixM suchthattheMahalanobis
distanceparametrizedbyM isbelowagiventhresholdlforsimilarpairsofpointsand
aboveagiventhresholdufordissimilarpairsofpoints.
LetD denotetheLogDetdivergencebetweenSPDmatricesdefinedas
ld
D (P,Q)=trace(PQ−1)−logdet(PQ−1)−n; P, Q∈S++. (1)
ld n
ITMLformulatestheMahalanobismatrixlearningasthefollowingoptimizationprob-
lem:
minimize D (M,M )+γD (diag(ζ),diag(ζ ))
ld 0 ld 0
M≻0,ζ
subjectto (x −x )⊤M(x −x )≤ζ , ∀(i,j)∈S (2)
i j i j c(i,j)
(x −x )⊤M(x −x )≥ζ , ∀(i,j)∈D,
i j i j c(i,j)
wherec(i,j)denotestheindexofthe(i,j)−thconstraint,ζ isthevectorofvariables
ζ , ζ is a vector whose components equal l for similarity constraints and u for
c(i,j) 0
dissimilarityconstraints,M isanSPDmatrixthatcapturesthepriorknowledgeabout
0
M,andγ isaparametercontrollingthetradeoffbetweensatisfyingtheconstraintsand
minimizing D (M,M ). This optimization problem can be solved efficiently using
ld 0
Bregmaniterations.Inthiswork,weusethepubliclyavailableITMLcodeprovidedby
theauthorsof[7].
ITML parameters: We need to specify the values for the following parameters
while using ITML: M , γ, l, u. We choose the constraint thresholds l and u as the
0
athandbthpercentilesoftheobserveddistributionofdistancesbetweenpairsofpoints
withinthetrainingdataset.Hence,theparametersfortheITMLalgorithmareM , γ, a
0
andb.
4 RavitejaVemulapalli,DavidW.Jacobs
Table1:SPDmatrixdistancesandtheirproperties
Triangle
Distance Formula Symmetric Geodesic
inequality
Frobenius kP1−P2kF Yes Yes No
Cholesky- kChol(P1)−Chol(P2)kF Yes Yes No
Frobenius[13]
J-divergence[12] 12qtrace(P1P2−1+P2P1−1)−2n Yes No No
Jensen-Bregman
LogDet Diver- qlogdet(cid:0)P1+2P2(cid:1)− 12logdet(P1P2) Yes No No
gence[11]
A[1f]fine-invariant klog(cid:16)P1−1/2P2P1−1/2(cid:17)kF Yes Yes Yes
Log-Frobenius klog(P1)−log(P2)kF Yes Yes Yes
[6]
Table2:SPDmatrixdistancesandtheirproperties
Distance Affine Scale Rotation Inversion
Distance
fromS+ invariance invariance invariance invariance
n
Frobenius Finite No No Yes No
Cholesky-Frobenius
Finite No No No No
[13]
J-divergence[12] Infinite Yes Yes Yes Yes
Jensen-Bregman
Infinite Yes Yes Yes Yes
LogDetDivergence[11]
Affine-invariant[1] Infinite Yes Yes Yes Yes
Log-Frobenius[6] Infinite No Yes Yes Yes
RiemannianMetricLearningforSPDMatrices 5
4 Log-Euclidean Riemannian metriclearning
The log-Euclideanframework [6] proposed by Arsigny et. al. defines a class of Rie-
mannianmetricscalledlog-Euclideanmetrics.Thegeodesicdistancesassociatedwith
log-Euclideanmetricsarecalledlog-Euclideandistances.Let⊙beanoperationonSPD
matricesdefinedasP ⊙P = exp(log(P )+log(P )).Wehavethefollowingresult
1 2 1 2
basedonthelog-Euclideanframeworkintroducedin[6]:
Result4.1:Anyinnerproducth, idefinedonT S++ = {log(P)|P ∈ S++} = S
I n n n
extended to the Lie group (S++,⊙) by left- or right- multiplication is a bi-invariant
n
Riemannian metric. The corresponding geodesic distance between P ∈ S++ and
1 n
P ∈S++ isgivenby
2 n
d(P ,P )=kmlog (P )−mlog (P )k =klog(P )−log(P )k , (3)
1 2 I 1 I 2 I 1 2 I
wherekk isthenorminducedbyh, i.Notethatheremlog istheinverse-exponential
I I
mapattheidentitymatrixwhichisequaltotheusualmatrixlogarithminthiscase.
Thesetofalln×nsymmetricmatricesformavectorspaceofdimensiond= n(n+1).
2
Letvec(P)denotethecolumnvectorformoftheuppertriangularpartofamatrixP.
Thisvec()operationprovidesaddimensionalvectorrepresentationforS .Leth, ibe
n
an innerproductdefinedonthevectorspaceS andM ∈ S++ bethecorresponding
n d
matrixofinnerproductsbetweenthedbasisvectorscorrespondingtovec()representa-
tion.Notethath, iisuniquelycharacterizedbyM.Thedistancebetweentwomatrices
P ∈S andP ∈S inducedbythisinnerproductisgivenby
1 n 2 n
⊤
d(P ,P )=(vec(P )−vec(P )) M(vec(P )−vec(P )). (4)
1 2 1 2 1 2
Result 4.2: Let M ∈ S++ , where d = n(n+1). Then, M defines a unique inner
d 2
product denoted by h, i on T S++ = {log(P) | P ∈ S++} = S . This inner
M I n n n
producth, i alsodefinesalog-EuclideanRiemannianmetricwhichcanbeobtained
M
bysimplyextendingh, i totheLiegroup(S++,⊙)byleft-orright-multiplication.
M n
Thecorrespondinglog-EuclideangeodesicdistancebetweenP ∈S++andP ∈S++
1 n 2 n
isgivenby
⊤
d (P ,P )=(vec(log(P ))−vec(log(P ))) M(vec(log(P ))−vec(log(P ))).
M 1 2 1 2 1 2
(5)
Theaboveresultfollowsdirectlyfromresult4.1.Result4.2saysthatanyMahalanobis
distance defined in the vector space {vec(log(P)) | P ∈ S++} is a geodesic dis-
n
tance on S++ and the corresponding Riemannian metric is uniquely defined by the
n
Mahalanobis matrix M. Hence, we can learn Riemannian metrics/geodesic distances
forS++ fromthedatabylearningMahalanobisdistancefunctionsinthevectorspace
n
{vec(log(P))|P ∈S++}.Table3summarizesourapproachforleaninggeodesicdis-
n
tancesonS++.Inthiswork,weuseITMLtechniqueforMahalanobisdistancelearning.
n
6 RavitejaVemulapalli,DavidW.Jacobs
Table3:AlgorithmforlearninggeodesicdistancesonS++
n
Input:{P ∈S++}N
i n i=1
for i=1toN
v =vec(log(P ))
i i
end
LearnaMahalanobisdistancefunctionusing{v }N .ThisgivesaMahalanobismatrix
i i=1
M ∈S++,whered= n(n+1).
d 2
Output:GeodesicdistancebetweenP andP :
1 2
(vec(log(P ))−vec(log(P )))⊤M(vec(log(P ))−vec(log(P ))).
1 2 1 2
5 Experiments
Inthesection,weevaluatetheperformanceoftheproposedRiemannianmetric/geodesic
distancelearningapproachontwoapplications:(i)FacematchingusingLabeledFaces
intheWild(LFW)datasetand(ii)Semi-supervisedclusteringusingETH80dataset.
5.1 FacematchingusingLFWfacedataset
Inthisexperimentouraimistopredictwhetheragivenpairoffaceimagescorrespond
tothesamepersonornot.
Dataset:TheLFWdataset[9]isacollectionoffacephotographsdesignedforstudying
the problemofunconstrainedfacerecognition.Thisdatasetconsistsof13233labeled
faceimagesof1680subjectscollectedfromtheweb.Thisdatasetconsistsoftwosub-
sets:
– Development subset: The development subset consists of 2200 training image
pairs, where 1100 are similar pairs and 1100 are dissimilar pairs, and 1000 test
image pairs, where 500 are similar pairs and 500 are dissimilar pairs. An image
pair is said to be similar if both the images correspond to the same person and
dissimilariftheycorrespondtodifferentpersons.
– Evaluationsubset:Theevaluationsubsetconsistsof3000similarimagepairsand
3000 dissimilar image pairs. It is further divided into 10 subsets each of which
consistsof300similarpairsand300dissimilarpairs.
Alltheimagepairsweregeneratedbyrandomlyselectingimagesfromthe13233im-
agesinthedataset.Thedevelopmentsubsetismeantformodelandparameterselection.
Theevaluationsubsetshouldbeusedonlyonceforfinaltrainingandtesting.Toavoid
overfitting,theimagepairsinthedevelopmentsubsetwerechosentobedifferentfrom
theimagepairsintheevaluationsubset.
Featureextraction We cropthefaceregionineachimageandresizeittoa64×64
image.Following[3],we converteachpixelin animageinto a 9-dimensionalfeature
vectorgivenby
⊤
∂W(x,y) ∂W(x,y) ∂2W(x,y) ∂2W(x,y)
x,y,R(x,y),G(x,y),B(x,y), , , , ,
(cid:20) (cid:12) ∂x (cid:12) (cid:12) ∂y (cid:12) (cid:12) ∂x2 (cid:12) (cid:12) ∂y2 (cid:12)(cid:21)
(cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12)
RiemannianMetricLearningforSPDMatrices 7
wherex, yarethecolumnandrowcoordinatesrespectively,R,GandB arethecolor
coordinatesandW isthegrayscaleimage.We usethe9×9covariancematrixofthe
featurevectorstorepresenttheimage.
Experimentalprotocol Followingthestandardexperimentalprotocolforthisdataset,
we use the development set for selecting the parameters of ITML and then use the
evaluationsetonlyonceforfinaltrainingandtesting.Followingstepssummarizeour
experimentalprocedure:
– Parameterselection:We trainthe ITMLalgorithmusingthe2200trainingpairs
ofthedevelopmentsubsetandthentestitonthe1000testpairsofthedevelopment
subset.WeselecttheITMLparametersthatgivethebesttestaccuracy.
– Finaltrainingandtesting:Theevaluationsetconsistsof10splitsandweperform
10-foldcross-validation.Ineachfold,weuse9splits(2700similarpairsand2700
dissimilarpairs)fortrainingITMLand1split(300similarpairsand300dissimilar
pairs) for testing. For training ITML, we use the parameters that were selected
in the previous step. Since our task is face matching, we need to threshold the
learneddistancefunction.Ineachfold,wefindthethresholdthatgivesbesttraining
accuracyandusethesamethresholdfortestimagepairs.
Comparative methods We comparethe performanceof the proposedlog-Euclidean
metriclearningapproachwiththefollowingapproaches:
– Directlyuseanyofthefollowingdistancesformatching:
• Frobenius,Cholesky-Frobenius,J-divergence,Jensen-BregmanLogDetdiver-
gence,Affine-invariantandLog-Frobenius.
– Use ITML directly with the covariance matrices by treating them as elements of
theEuclideanspaceofsymmetricmatrices.
– UseITMLwiththelowertriangularmatrixobtainedbyCholeskydecomposition.
Inallthesemethodsthedistancethresholdisobtainedineachfoldindependentlybased
onthetrainingdata.
Parameters ThefollowingparametervalueswereusedforITML:
– M =I, γ =103.5, a=5, b=95.
0
Theseparameterswereselectedusingthedevelopmentsubsetofthedataset.
Results Tables4and5summarizethepredictionresultsforvariousapproachesonthe
LFWdataset.Wecandrawthefollowingconclusionsfromtheseresults:
– TheproposedRiemannianmetric/geodesicdistancelearningapproachoutperforms
theotherapproachesforcomparingcovariancematrices.
– The log-Euclideangeodesic distance learned from the data performsmuch better
thanthestandardlog-Frobeniusdistance.
– Distance learning with original covariance matrices or Cholesky decompositions
performspoorlycomparedtodistancelearninginthelogarithmdomain.
8 RavitejaVemulapalli,DavidW.Jacobs
Table4:PredictionaccuracyonLFWdatasetusingvariousSPDmatrixdistances
Cholesky- Jenson-Bregman Affine
Frobenius Log-Frobenius J-divergence
Frobenius LogDetdivergence invariant
53.77 56.62 60.43 60.92 61.62 61.15
Table5:PredictionaccuracyonLFWdatasetusingdistancelearning
Covariancematrices Choleskydecompositions Log-Euclidean
Frobenius ITML ITMLgain Frobenius ITML ITMLgain Frobenius ITML ITMLgain
53.77 57.58 3.81 56.62 63.53 6.91 60.43 69.37 8.94
5.2 Semi-supervisedclusteringusingETH80objectdataset
Inthisexperiment,weareinterestedinclusteringtheimagesintheETH80datasetinto
differentobjectcategories.
Dataset TheETH80objectdataset[10]consistsof256×256imagesof8objectcate-
gorieswitheachcategoryincluding10differentobjectinstances.Eachobjectinstance
has41imagescapturedunderdifferentviews.So,eachobjectcategoryhas410images
resultinginatotalof3280images.
Feature extraction We convert each pixel in an image into a 9-dimensionalfeature
vectorgivenby
⊤
∂W(x,y) ∂W(x,y) ∂2W(x,y) ∂2W(x,y)
x,y,R(x,y),G(x,y),B(x,y), , , , ,
(cid:20) (cid:12) ∂x (cid:12) (cid:12) ∂y (cid:12) (cid:12) ∂x2 (cid:12) (cid:12) ∂y2 (cid:12)(cid:21)
(cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12)
wherex, yarethecolumnandrow(cid:12) coordinate(cid:12)sr(cid:12)espectively(cid:12),R(cid:12),GandB a(cid:12)re(cid:12)thecolor (cid:12)
coordinatesandW isthegrayscaleimage.Wecomputethe9×9covariancematrixof
thefeaturevectorsovertheentireimageanduseittorepresenttheimage.
Experimentalprotocolandparameters Foreveryobjectcategory,werandomlyse-
lect4imagesfromeachinstancefortraining.Hence,weuse40samplesfromeachob-
jectcategoryfortraining,resultinginatotalof320trainingimages.Fromeachpairof
trainingimages,wegenerateeitherasimilarityconstraintoradissimilarityconstraints
basedontheircategorylabels.WeuseallsuchconstraintsinlearningtheMahalanobis
distancefunction.OncewelearntheMahalanobisdistancefunction,weuseitforclus-
teringtheentiredatasetof3280images.
We repeatthe aboveprocedure5 times and reportthe averageclustering accuracy.In
eachrun,weselectthevalueofITMLparameterγ usingtwofoldcross-validationon
the trainingdata. We use the followingvaluesforother ITML parametersin all the 5
runs:M =I, a=5, b=95.
0
We use K-meansalgorithmfor clustering.To handlethelocal-optimumissue, we run
RiemannianMetricLearningforSPDMatrices 9
K-meanswith20differentrandominitializationsandselecttheclusteringresultcorre-
spondingtotheminimumK-meanscostvalue.
Comparative methods We comparethe performanceof the proposedlog-Euclidean
metriclearningapproachwiththefollowingapproaches:
– Unsupervised: Directly perform K-means clustering using any of the following
distances:Frobenius,Cholesky-FrobeniusandLog-Frobenius.
– Use ITML directly with the covariance matrices by treating them as elements of
theEuclideanspaceofsymmetricmatrices.
– UseITMLwiththelowertriangularmatrixobtainedbyCholeskydecomposition.
Computationofmeandoesn’thaveaclosedformsolutioninthecaseofJ-divergenceor
Jensen-BregmanLogDetdivergenceorAffine-invariantdistance.Hence,weneedtouse
someoptimizationprocedureforcomputingthemean.ThismakesK-meansalgorithm
highlycomputational.Hence,wedonotusethesedistancesforcomparisoninthiswork.
Results Table6summarizestheclusteringresultsforvariousapproachesontheETH80
dataset.Wecandrawthefollowingconclusionsfromtheseresults:
– The proposed Riemannian metric/geodesic distance learning approach performs
betterthanotherapproachesforclusteringSPDmatrices.
– The log-Euclideangeodesic distance learned from the data performsmuch better
thanthestandardlog-Frobeniusdistance.
– Distance learning with original covariance matrices or Cholesky decompositions
performspoorlycomparedtodistancelearninginthelogarithmdomain.
6 Conclusion
Inthiswork,wehaveexploredtheideaofdata-drivenRiemannianmetricsorgeodesic
distances.Basedonthelog-Euclideanframework[6],wehaveshownhowgeodesicdis-
tancefunctionscanbelearnedforS++bysimplylearningMahalanobisdistancefunc-
n
tions in the logarithmdomain.We have conductedexperimentsusing face and object
data sets. Thefacematchingandsemi-supervisedobjectcategorizationresultsclearly
showthatthelearnedlog-Euclideangeodesicdistanceperformsmuchbetterthanother
distances.
Table6:ClusteringaccuracyonETH80dataset
Covariancematrices Choleskydecompositions Log-Euclidean
Frobenius ITML ITMLgain Frobenius ITML ITMLgain Frobenius ITML ITMLgain
35.58 70.50 34.92 51.13 70.36 19.24 55.70 73.79 18.09
10 RavitejaVemulapalli,DavidW.Jacobs
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