a constrained, weighted - 1 minimization approach for ... · introduction recently, there has been...

AConstrained,Weighted-ℓ1MinimizationApproachforJointDiscoveryofHeterogeneousNeuralConnectivityGraphs

Chandan Singh1;Beilun Wang2;Yanjun Qi21UniversityofCalifornia,Berkeley,2UniversityofVirginia

1. DiMartino,A.;Yan,C.-G.;Li,Q.;Denio,E.;Castellanos,F.X.;Alaerts,K.;Anderson,J.S.;Assaf,M.;Bookheimer,S.Y.;Dapretto,M.;etal.2014.Theautismbrainimagingdataexchange:towardsalarge-scaleevaluationoftheintrinsicbrainarchitectureinautism.Molecularpsychiatry19(6):659–667.

2. PreprocessingconductedbyABIDEpreprocessedconnectomesproject:Craddock,C.2014.Preprocessedconnectomesproject:opensharingofpreprocessedneuroimagingdataandderivatives.In61stAnnualMeeting.AACAP.

3. Connectomesvisualizedwithnilearn:Abraham,A.;Pedregosa,F.;Eickenberg,M.;Gervais,P.;Muller,A.;Kossaifi,J.;Gramfort,A.;Thirion,B.;andVaroquaux,G.2014.Machinelearningforneuroimagingwithscikit-learn.arXiv preprintarXiv:1412.3919.

4. Cai,T.;Liu,W.;andLuo,X.2011.Aconstrained1minimiza- tion approachtosparseprecisionmatrixestimation.JournaloftheAmericanStatisticalAssociation106(494):594–607.

5. Friedman,J.;Hastie,T.;andTibshirani,R.2008.Sparsein- versecovarianceestimationwiththegraphicallasso.Biostatistics9(3):432–441.

6. Wang,B.;Singh,R.;andQi,Y.2016.Aconstrainedl1minimiza- tion approachforestimatingmultiplesparsegaussian ornonpara-

normalgraphicalmodels.arXiv preprintarXiv:1605.03468.7. Danaher,P.;Wang,P.;andWitten,D.M.2014.Thejointgraphicallassoforinversecovarianceestimationacrossmultipleclasses.

JournaloftheRoyalStatisticalSociety:SeriesB(Statisti- cal Methodology)76(2):373–397.8. Chiquet,J.;Grandvalet,Y.;andAmbroise,C.2011.Inferringmul- tiple graphicalstructures.StatisticsandComputing21(4):537–553.9. Dosenbach,N.U.;Nardos,B.;Cohen,A.L.;Fair,D.A.;Power,J.D.;Church,J.A.;Nelson,S.M.;Wig,G.S.;Vogel,A.C.;Lessov-

Schlaggar,C.N.;etal.2010.Predictionofindividualbrainmaturityusingfmri.Science329(5997):1358–1361.

References

Determiningfunctionalbrainconnectivityiscrucialtounderstandingthebrainandneuraldifferencesunderlyingdisorders,suchasautism.RecentstudieshaveusedGaussiangraphicalmodelstolearnbrainconnectivityviastatisticaldependenciesacrossbrainregionsfromneuroimaging.However,previousstudiesoftenfailtoproperlyincorporatepriorstailoredtoneuroscience,suchaspreferringshorterconnections.Toremedythisproblem,thepaperhereintroducesanovel,weighted-ℓ1,multi-taskgraphicalmodel(W-SIMULE).

W-SIMULEelegantlyincorporatesaflexibleprior,alongwithaparallelizableformulation.Additionally,W-SIMULEextendstheoften-usedGaussianassumption,leadingtoconsiderableperformanceincreases inapplicationstofMRIdata.Duetoitselegantdomainadaptivity,W-SIMULEcanbereadilyappliedtovariousdatatypestoeffectivelyestimateconnectivity.

AbstractTheproblemofdeterminingfunctionalbrainconnectivityconcernsusingthecovariancematrix(Σ)tocalculatetheprecisionmatrixΩ),whichrepresentsconditionalcorrelationsbetweenbrainareas.Todothis,W-SIMULEusesfourproperties:

1. Sparsity2. Multi-tasklearningwithK groups3. ApriormatrixofpositiveweightsW4. Anonparanormalassumption

CombiningtheseelementsyieldsthenovelformulationofW-SIMULE:

Ω$%('), … , Ω$%

+ , Ω,- =/argmin 𝑊 ⋅ Ω$9

'+ 𝜖𝐾 𝑊 ⋅ Ω- '

�

9

Subjectto: ΣG9 Ω$

9 + Ω- − 𝐼J≤ 𝜆, 𝑖 = 1: 𝐾

W-SIMULEhasthreehyperparameters:

1. W– enforcesadifferentpriororchangeshowstrictlyitisenforced2. λ - controlsthetotalsparsityoftheresultingprecisionmatrices3. ε – controlshowstrictlythegrouppenaltyisimposed

Method:W-SIMULE

W-SIMULEgreatpotentialforfutureapplications.Asbrain-imagingdatasetsbecomemorecomplexandincludemorestructuraldatacoupledwithfunctionaldata,W-SIMULEwillbecomeincreasinglyimportanttoneuroscience.Thisisespeciallytrueforstudieswithsmallsamplesizes,suchastask-specificstudies,whichrequirestrongpriorsandmulti-tasklearninginordertorobustlydetermineconnectivity.AsthespatialresolutionoffMRIincreases,spatialpenalizationwillbecomemoreimportantinconstructingaccurateROIsandbrainconnections.

ManyproblemsoutsideofneurosciencecanbenefitfromW-SIMULE;itcanutilizediversepriorstofindconditionalindependencebetweennodesinanymulti-tasksetting.Thus,W-SIMULEcanbereadilyappliedtogene-networkestimation,imageprocessing(wherephysicaldistancecouldbeusedasapriorinimages),andmanyotherproblemsthatcurrentlyutilizeGaussiangraphicalmodels.

Conclusions/FutureWork

IntroductionRecently,therehasbeengreatinterestinmappingtheinteractionsbetweenbrainregions,afieldknownasfunctionalconnectomics.Wefocusontheimportantproblemofestimatingbrainconnectivityformorethanonegroup(i.e.adiseasegroupandacontrolgroup).Thisstudy'smaincontributionisthenovelformulationofW-SIMULE,whicharisesnaturallyfrombrain-imagingdata.W-SIMULEisaweighted-ℓ1,multi-taskgraphicalmodelwhichrobustlyestimatestheprecisionforeachgroup.Itsmainadvantagesare:

1. Effectiveness. ityieldsaccurateconnectivityintermsoflog-likelihoodandclassificationaccuracyontheABIDEresting-statefMRIdataset1

2. Domainadaptivity.itelegantlyenforcesapriorbasedontheproblemathandandcanovercometheoften-incorrectGaussianassumptionbyusingnonparanormality

3. Interpretability.itcalculatesaconnectomeforeachgroupwhichcanbetunedtothedesiredsparsitylevelandisparticularlyeffectiveatlowsparsitylevels

4. Efficiency. theformulationiscolumn-wiseparallelizableandquicklysolvable

Enforcingtheprior.9 W-SIMULEeffectivelyenforcestheprior(Fig3A).Asthedist priorisraisedtoahigherpower(therebyincreasingthespreadofthepriorweights),thepriorismorestrictlyenforced,resultinginaloweraverageedgelengthateverysparsitylevel.The“optimal”lineshowsthelowestpossibleaverageedgelengthasafunctionofgraphsparsity.

Log-likelihood. Connectomesaregeneratedforvarioussparsitylevelsandtheirresultinglog-likelihoodsareplottedinFig3B.W-SIMULE(dist2 prior)outperformsalloftherelevantbaselines,especiallyforverysparseconnectomes,whicharethemostinterpretable.

ClassificationAccuracy. Table1displaysthemaximumaccuracyachievedforeachbaseline,aftersweepingoverhyperparameters.W-SIMULE(W=anatomical)yieldsaclassificationaccuracyof58.62%.

Results

Fig1. Datapipelineforconstructingafunctionalconnectome.2

1 .2 .1 .3 .4.3 1 .8 .3 .4.6 .5 1 .7 .2.3 .7 .2 1 .7.7 .5 .2 .4 1

BrainsscannedwithfMRI

160signalsextracted2 Correlationsbetweensignals Connectome3

Prep

rocessed

fMRIsignal

Fig2. ToyexampledepictingW-SIMULE.Leftshowspotentialedgespresentinthedataandrightshowslearnededges.Longedges(red)arespatiallypenalizedanddiscarded,edgesthatdifferbetweengroups(blue)arelearnedindividually,andedgessharedbetweengroups(black)arelearnedin𝛀,𝑺.

W-SIMULE CLIME4 GLASSO5 SIMULE6 JGL(fused)7 SIMONE8

58.62 46.55 53.71 57.96 56.90 53.71

Table1. ClassificationaccuracyobtainedontheABIDEdatasetusingvariousmethods.

Fig4.SparseconnectomegeneratedbyW-SIMULEwiththeanatomicalprior.Theautismgraph,controlgraph,andtheirdifferenceareshown.

Fig3.Connectivityresults.A showsthatW-SIMULEeffectivelyenforcespriorsthatpenalizeslongedges.B&C showthatW-SIMULEcanmaximizethelog-likelihoodmoreeffectivelythanothermethods..

A B C

Log-Likelihoo

d