Co-selling recommendations from an eCommerce...

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Co-sellingrecommendationsfromaneCommercedataset.PawanRewari–SUID01421653–dadrewari@gmail.com

CS224W–ProjectMilestoneUpdate1.0Introduction/Motivation/ProblemDefinitionAverysimplecustomeruse-caseofeCommerceisasfollows.LetssayJanewantstobuyaspecificproduct(aKielty2personbackpackingtent)oroneofacategoryofitems(atent).TypicallyshewillgototheweboraneCommercesiteanddoasearch.Thenshemayresearchtheproductandafterseveralvisitseitherthesamedayoroverseveraldaysorweekseventuallyconvergetoapurchasingdecision.Eachsuchinteractionisanopportunityforthesellertoshowcasethespecificproductsheislookingforandshowotherproductsthatsheislikelytobeinterestedin.Fromaretailerperspectivetheywanttokeeptheproductsshowcasedtothecustomerrelevanttotheirpurchaseinterest,andtheywanttoprovideaninterestingvarietyofproductstoshowcase.TheoverallproblemwesetouttoresearchishowcanoneprovideanewandnovelapproachtosellingmoreproductsataneCommercesite.Therealworldapplicationisadatascienceteam,asconsultants,whoareapproachedtoprovidestrategiestosellmoreproductsandgivenasliceofhistoricaldata–andaskedtoworkwithwhattheyaregiven.WhilethisprojectisfocusedonaneCommercedatasetfromAmazon(co-sellingproducts)wewouldliketoextrapolatethefindingstoinformanyeCommerceretaileronthevalueofsuchnetworkanddataanalysis.Specifically,ifwearegivenapartialdatasettoworkwith,asisthecasewithourdatasetandnocustomerinformationbeyondthisaggregateddata.Wehavebeenprovidedwithadatasetthatincludes:

- 548552products(including5868discontinuedproducts)- foreachproductrelevantinformationincludes:

o Id:nodesorverticesnumberedfrom0to548551o ASIN:AmazonStandardIdentificationNumberforaproducto Group:Book,Video,DVD,Music(theprimarygroups)o SalesRank:computedbyAmazonasafunctionofhistoricalsales,basedonnumber

ofproductssold;alowerranksignifiesahighersellingproducto Similar:productsthathaveco-sold(Similar==Co-Selling)o Categories:thecategoriesthisproductbelongsto,canbe10deep

Theinitialexploratoryanalysisofthedatasetrevealedthattherewereatmost5co-sellingrecommendationsforanyproduct(the“Similar”products)andthiswasonlyfor60%oftheproductsinthedataset,fortherestnoneorfewerrecommendations(seeFigure1nextpage).Nowasstatedintheintroductoryusecase,ifyoulookatatypicalcustomer’sbehavior,theyarelikelytovisitaneCommercewebsiteseveraltimesbeforetheyconsummateapurchase.Eachsuchvisitisanopportunitytoselltheproductandrecommendsimilaronesandyouwanttohaveavarietyofproductstorecommend,onetofiveisnotenoughweneedmore,manymore,Thereinliesthecoreproblemstatementthatweaddressinthispaper.Howdoyoutoprovideanaddedsetofrecommendationsthatareuseful,relevantandonaveragetaketheretailertoabetterplacethantheystartedoutwith.

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Soifwelookatthebasicdistributionofrecommended/similarproductsprovidedbytheinitialdatasetwehave:Discontinued 5868 1%0Similar 163591 30%1Similar 11071 2%2Similar 10808 2%3Similar 10275 2%4Similar 9482 2%5Similar 337457 62%Total 548552 100%Sotheproblemstatementbecameveryclearwiththreeobjectives:

1. providearecommendationforalargernumberof“Similar”productssoyoucanprovidefreshdataonmultiplevisitsbytheshopper(metric–morerecommendedproducts)

2. increasethe“relevance”oftheaddedsimilarproductswithstronger“categoricalsimilarity”–explainedindetailinSection3,brieflyitisthedepthofsimilarcategoriesfor2products(metric–raisetheavgcategoricalsimilarityofrecommendedproducts)

3. improvetheaverageSalesRankoftherecommendedset(metric–lowertheavgSalesRankofrecommendedproducts)

This translates to 3 measures that we call the 3 CoSellingMetrics. We create a baseline with the initial dataset and try toimproveuponeachofthesemeasures.Intherealworldwewouldrunactualexperimentstoseeiftheserecommendationsactuallyimprovesalesandtheniterativelyimprovetherecommendations,inourcasesincewecan’tdorealtrialswearemeasuringperformancerelativetothebaselineestablishedbythe3CoSellingMetrics.Goingbacktothesimpleuse-case,wewouldliketorecommend5productsforCoSaleateveryvisitandthenrecommendotherproductsonnewvisits,butwewanttodothisfromalimitedsetofSimilarproductsthatwedrawfrom,sowhiletheuserseesdifferentproductsrecommended,theyalsoseethesameonesrecyclebackonsubsequentvisits–onestheymaybeinterestedin.Sowesetareasonabletargetfortherecommendedproductstobe~15,thisisrelativelyarbitrarybutmakessensefortheusecasewehavehere.

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2.0RelatedWorkAsstatedintheReferenceswedidaliteraturereviewofseveralpapersonCollaborativeFiltering,ClusteringMethodsandItem-to-ItemCollaborativeFiltering.Hereisasummary.AtraditionalcollaborativefilteringalgorithmrepresentsacustomerasanN-dimensionalvectorofitems,whereNisthenumberofdistinctcatalogitems.Thecomponentsofthevectorarepositiveforpurchasedorpositivelyrateditemsandnegativefornegativelyrateditems.Tocompensateforbest-sellingitems,thealgorithmtypicallymultipliesthevectorcomponentsbytheinversefrequency(theinverseofthenumberofcustomerswhohavepurchasedorratedtheitem),makinglesswell-knownitemsmorerelevant.Formostcustomers,thisvectorisextremelysparse.Thealgorithmgeneratesrecommendationsbasedonafewcustomerswhoaremostsimilartotheuser.Itcanmeasurethesimilarityoftwocustomers,AandB,invariousways;acommonmethodistomeasurethecosineoftheanglebetweenthetwovectors.Thealgorithmcanselectrecommendationsfromthesimilarcustomers’itemsusingvariousmethodsaswell,acommontechniqueistorankeachitemaccordingtohowmanysimilarcustomerspurchasedit.Inclustermodelstofindcustomerswhoaresimilartotheuser,dividethecustomerbaseintomanysegmentsandtreatthetaskasaclassificationproblem.Thealgorithm’sgoalistoassigntheusertothesegmentcontainingthemostsimilarcustomers.Itthenusesthepurchasesandratingsofthecustomersinthesegmenttogeneraterecommendations.Becauseoptimalclusteringoverlargedatasetsisimpractical,mostapplicationsusevariousformsofgreedyclustergeneration.Thesealgorithmstypicallystartwithaninitialsetofsegments,whichoftencontainonerandomlyselectedcustomereach.Theythenrepeatedlymatchcustomerstotheexistingsegments,usuallywithsomeprovisionforcreatingnewormergingexistingsegments.Forverylargedatasets—especiallythosewithhighdimensionality—samplingordimensionalityreductionisalsonecessary.Clustermodelshavebetteronlinescalabilityandperformancethancollaborativefilteringbecausetheycomparetheusertoacontrollednumberofsegmentsratherthantheentirecustomerbase.However,recommendationqualityislow.Ratherthanmatchingtheusertosimilarcustomers,Amazon’sitem-to-itemcollaborativefilteringmatcheseachoftheuser’spurchasedandrateditemstosimilaritems,thencombinesthosesimilaritemsintoarecommendationlist.Todeterminethemost-similarmatchforagivenitem,thealgorithmbuildsasimilar-itemstablebyfindingitemsthatcustomerstendtopurchasetogether.Theybuildaproduct-to-productmatrixbyiteratingthroughallitempairsandcomputingasimilaritymetricforeachpairbasedonthecosineofthetwovectors.Givenasimilar-itemstable,thealgorithmfindsitemssimilartoeachoftheuser’spurchasesandratings,aggregatesthoseitems,andthenrecommendsthemostpopularorcorrelateditems.Thiscomputationdependsonlyonthenumberofitemstheuserpurchasedorrated.AsstatedinSection1,withourdatasetwehaveinformationonproducts,co-sellingrelations&categories,butnotoncustomers.Thismayverywellbethecaseinarealworldsituationwhereyouareconsultingforacustomerwhoprovidesyouwithalimiteddataset.Sincewearenotprovidedwithcustomerpurchasedata,onlyaggregatedata,wecouldlearnfrombutnotdirectlyapplytheaboveideas.WecameupwithanapproachthatallowedustomakegoodrecommendationsbyimprovingonthethreeCoSellingMetricsstatedearlier.

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3.1ApproachWestartwithaninitialsparsesetofrecommendations(representedinG1),atreeofcategoriesandsub-categories(G2)andusethesetocomeupwithadensersetofrecommendations(G3)asshown.The3Graphs(Figure1)relevanttoourapproach&analysisare:

• G1:G(V,E),eachVertexisaproductandeachEdgerepresentsaco-sellingrelationship.Thisisanundirectedgraph(ifIsellwithyou,yousellwithme).EachVertexhasanASIN,Group,SalesRank.EdgestoallSimilar/co-sellingproducts.

• G2:(V,E)isthecategorytree.TheVerticesareallthecategories&sub-categories.Therootnodeisproduct,nextlevelisBooks,Music,Video,DVD,theirsub-categoriesatthenextlevelandsoon.TheEdgesindicatesacategorytosub-categoryrelationship.Eachproductthenbelongstomultiplecategorypathsasillustratedonthenextpage.

• G3:G(V,E)isbuiltontopofG1andusestheinformationfromG2,itcontainsasignificantnumberofaddededgesrepresentingnewlyrecommendedco-sellingrelationshipsbyourrecommendationenginedescribedin3.2.

MetricsforinitialGraphG1:G1:Nodes542684,Edges987903G1:AverageDegree:1.82G1:MaxWcc:Nodes334852,Edges925823G1:Diameter:43G1:AverageClusteringCoefficient:0.274496Figure1:G1,G2&G3

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Baselineandwhatwesetouttomeasure.ThethreeCoSellingMetricsarebasedon:1:AvgRecommendedProducts–thenumberofSimilarproducts(avgDegree) Thisissimplythe#edges(degree)ofeachnode,averagedacrossallnodes2:AvgCategoricalSimilarity–theCategoricalSimilarityaverageoftherecommendedproducts. Thisisexplainedbelow3:AvgSalesRank–averageSalesRankofthesimilarproductsthatarerecommended Eachproduct(node)hasaSalesRank,soweaveragetheSalesRankofallitneighborsTheobjectiveistoincrease1(Co-Selling)&2(relevance)anddecrease3(lowSalesRankisbetter).Figure2:PartialtreeofG2:_________CategorytoSub-CategoryEdges_________Product1belongsto3categorypaths_________Product2belongsto3categorypathsProduct1&Product2MaxPathintersectionatLevel4CategoricalSimilarity=4[Note–eachcategorypathdoesnotneedtoreachaleafofthetree]

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AverageCategoricalSimilarity:Astherealexamplebelowshowseachproduct(ASIN:0486220125)belongstoanumberofcategorypaths,eachproductitissimilarto(ASIN:0486401960&ASIN:0486229076&ASIN:0714840343)alsobelongstoanumberofcategorypaths.WelookforthehighestCategoryLevelthatiscommonbetweenthe“Subject”andtheproductsthatis“SimilarTo”…wethenaveragethisacrossall“Similarproducts”.

ASIN:0486220125 title:HowtheOtherHalfLives:StudiesAmongtheTenementsofNewYork

Similarto:=Co-PurchasingRelationship

ASIN:0486401960

ASIN:0486229076 ASIN:0714840343

Categorical

CategoryPathstheproduct(ASIN)islistedin

Similarity

CategoryLevels 1 2 3 4 5 6 7

Books[283155] Subjects[1000]Arts&Photography[1] Photography[2020] PhotoEssays[2082]

Subject Books[283155] Subjects[1000] History[9] Americas[4808] UnitedStates[4853] General[4870]

ASIN:0486220125 Books[283155] Subjects[1000] History[9] Jewish[4992] General[4993]

Books[283155] Subjects[1000] Nonfiction[53] SocialSciences[11232] Sociology[11288] Urban[11296]

[172282] Categories[493964]Camera&Photo[502394]

PhotographyBooks[733540] PhotoEssays[733676]

SimilartoSubject 6 Books[283155] Subjects[1000] History[9] Americas[4808] UnitedStates[4853] General[4870]

ASIN:0486401960 3 Books[283155] Subjects[1000] Nonfiction[53] CurrentEvents[10546] Poverty[10576] General[10578]

6 Books[283155] Subjects[1000] Nonfiction[53] SocialSciences[11232] Sociology[11288] Urban[11296]

4 Books[283155] Subjects[1000]

Arts&Photography[1] Photography[2020]| History[2032]

SimilartoSubject 4 Books[283155] Subjects[1000]Arts&Photography[1] Photography[2020] Travel[2087]

UnitedStates[2099] General[2100]

ASIN:0486229076 4 Books[283155] Subjects[1000]Arts&Photography[1] Photography[2020] Travel[2087]

UnitedStates[2099]

MidAtlantic[2101]

5 Books[283155] Subjects[1000] History[9] Americas[4808] UnitedStates[4853] State&Local[4872]

4 [172282] Categories[493964]Camera&Photo[502394]

PhotographyBooks[733540] Travel[733684]

UnitedStates[733708] General[733710]

4 [172282] Categories[493964]Camera&Photo[502394]

PhotographyBooks[733540] Travel[733684]

UnitedStates[733708]

MidAtlantic[733712]

SimilartoSubject 4 Books[283155] Subjects[1000]

Arts&Photography[1] Photography[2020]

Photographers,A-Z[2033]| General[2050]

ASIN:0714840343 5 Books[283155] Subjects[1000]Arts&Photography[1] Photography[2020] PhotoEssays[2082]

4 [172282] Categories[493964]Camera&Photo[502394]

PhotographyBooks[733540]

Photographers,A-Z[733566] General[733568]

5 [172282] Categories[493964]Camera&Photo[502394]

PhotographyBooks[733540] PhotoEssays[733676]

Toillustratethisindetail.TheSubject(ASIN:…0125)islistedin5categoryPaths.Ithasaco-sellingrelationship(Similarto)withASIN:…1960whichislistedin3CategoryPaths,withASIN:…9076whichislistedin6CategoryPaths,withASIN:…0343whichislistedin4CategoryPaths.IfIexaminethefirstCategoryPathofeachofthese,theyintersectwiththeCategoryPathsoftheSubjectatLevels6,4and4respectively.Wecontinuethisprocessandcomputeallthepathintersections(inYellow)andtaketheMaxCategoricalSimilarityineachcase(inGreen).AverageCategoricalSimilarity[fornodewithASIN:…0125]=Averageof(max)SimilaritywithASINs:196090760343=(6+5+5)/3=5.33

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BASELINEMetricsforG1:

OriginalDataset

WholeGraph AvgRecommendedProducts 1.82 AvgCategoricalSimilarity 3.25 AvgSalesRank 81729

Subset-AllProductswith1ormoreco-sellingproducts AvgRecommendedProducts 2.69 AvgCategoricalSimilarity 4.81 AvgSalesRank 67662Books

AvgRecommendedProducts 2.75 AvgCategoricalSimilarity 4.85 AvgSalesRank 84674Music

AvgRecommendedProducts 2.43 AvgCategoricalSimilarity 4.92 AvgSalesRank 26739Video

AvgRecommendedProducts 1.95 AvgCategoricalSimilarity 3.56 AvgSalesRank 7196DVD

AvgRecommendedProducts 3.43 AvgCategoricalSimilarity 5.03 AvgSalesRank 8430

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3.2Model/Algorithm/Method- formaldescriptionofanyalgorithmsused

Algorithmfollowed:

CreateG1(V,E):ReadtheinitialdatasetandeachVertexisaproductandeachEdgerepresentsaco-sellingrelationshipforeachVertex–ASIN,Group,SalesRank;EdgestoSimilarco-sellingproductsCreateG2(V,E):acategorytreewithsub-categoriesateachsub-level.Mapeachproducttothecategorypathsitbelongsto(typically3-7paths).Thepathsareprovidedbythedataset.Compute:AvgRecommendedProductsisthe#edges(degree)ofeachnode,averagedacrossallnodesAvgSalesRankistheaverageoftheSalesRankofallofitsneighbors.CategoricalSimilarityforany2productsisthehighestCategoryLeveltheyhaveincommonTheAvgCategoricalSimilarityisaverageCategoricalSimilarityacrossalltheproductsthatareinaco-sellingrelationshipwithaspecificproduct.CreateG3(V,E):StartwithG1.Foreachproduct–developalistofadditional“similar”productsbylookingatoneswiththehighestCategoricalSimilaritywiththeproduct,andamongsttheoneswiththesameCategoricalSimilarityhaveapreferenceforproductswithalowerSalesRank,stopafter~15orso(a“reasonable”targetweestablishedinSection1).AddtheseedgestoG3.Toidentifytheadditionalproductstocreateedgesto:(a) Forproducts(nodes)whichhaveoneormore“similar”productswefollowthepathofthe

neighborsandtheirneighbors…andevaluatetheonestoaddbasedonhigherCategoricalSimilarityandlowerSalesRank.

(b) Forallproductsinaddition,andforoneswithZERO“similar”productsthisistheonlywaytoaddmorerecommendations,lookforproductsthatareinthesamecategoriesandaddbasedonhigherCategoricalSimilarityandlowerSalesRank.

Re-ComputeAvgRecommendedProducts,AvgCategoricalSimilarity&AvgSalesRankandcomparewithinitialresults.Ideallywenowshouldhaveamuchdensergraphofco-sellingrelationshipswithmorerelevanceandlowerSalesRank.

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4.0ResultsandFindingsSoifwelookattheexamplewestartedwith,thenewsetof“similar”productsare:ASIN:0486220125 title:HowtheOtherHalfLives:StudiesAmongtheTenementsofNewYork

Newsetof"similar"products.ASIN:0486401960 title:TheBattlewiththeSlumASIN:0486229076 title:OldNewYorkinEarlyPhotographsASIN:0714840343 title:JacobRiis(55)ASIN:0451628810 title:TheFederalistPapersASIN:0395914787 title:AmericanVintage:TheRiseofAmericanWineASIN:1841762121 title:MatchlockMusketeer1588-1688(Warrior43)ASIN:0609608363 title:WondrousContrivances:TechnologyattheThresholdASIN:0807120820 title:Freedom'sLawmakers:ADirectoryofBlackOfficeholdersDuringReconstructionASIN:0060004436 title:PowerPlays:WinorLose--HowHistory'sGreatPoliticalLeadersPlaytheGameASIN:0807841625 title:TheEnduringSouth:SubculturalPersistenceinMassSocietyASIN:0805072845 title:ProjectOrion:TheTrueStoryoftheAtomicSpaceshipASIN:0811717003 title:StreetWithoutJoyASIN:0813911311 title:TheReligiousLifeofThomasJeffersonASIN:0801486955 title:GoingNative:IndiansintheAmericanCulturalImaginationASIN:0029024803 title:AnEconomicInterpretationoftheConstitutionofTheUnitedStatesFirst,justtakingacasuallookatthislist,itlookslikeareasonablesetofrecommendedproducts.Manyareinthethemeoftheoriginalproduct,andseveraltakeyoutonew,butrelatedplaces.SoeverytimeauservisitstheeCommercesitetopurchasetheywillbeshownasubsetof“peoplewholookedatthisalsolookedatthis”,“peoplewhopurchasedthis,alsopurchasedthis”.AswelookatthetransformationfromG1toG3,forthisONEproductwehavethenewmetrics:

• startingwith3(pg5)wenowhave15“similar”products(better)• theaveragecategoricalsimilaritygoesfrom5.33to5.72(better)• theaverageSalesRankfrom209144to299227(worse)

Fortheoverallgraphofsimilar/co-sellingproducts,weclearlyhaveatransformationtoamuchdensergraphwithasignificantlylargernumberofrecommendationsperproduct.MetricsforinitialGraphG1: MetricsfornewGraphG3:G1:Nodes542684,Edges987903 G3:Nodes542684,Edges7352419G1:AverageDegree:1.82 G3:AverageDegree:13.55G1:MaxWcc:Nodes334852,Edges925823 G3:MaxWcc:Nodes524997,Edges6426637G1:Diameter:43 G3:Diameter:11G1:AverageClusteringCoefficient:0.274 G1:AverageClusteringCoefficient:0.569Nowifwelookintothedetails,itclearlyshowsthatwehavesignificantlyraisedthenumberofrecommendedproductswhilesimultaneouslyimprovingaveragecategoricalsimilarity(relevance)andaverageSalesRank(rankbasedonhistoricalsales)oftherecommendedproducts.This

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involvedsignificantprogrammingwithmuchnewlearning,theresultsaresolidandwehaveconfidenceinthefindings.NEWMETRICS:

OriginalDataset

NewRecommendations Improvement

WholeGraph

AvgRecommendedProducts 1.82 13.55 7.4 X AvgCategoricalSimilarity 3.25 5.60 72.0% AvgSalesRank 81729 24343 70.2%

Graphsubset-AllProductswith1ormoreco-sellingproducts

AvgRecommendedProducts 2.69 13.97 5.2 X AvgCategoricalSimilarity 4.81 5.77 19.9% AvgSalesRank 67662 21654 68.0% Books

AvgRecommendedProducts 2.75 14.00 5.1 X AvgCategoricalSimilarity 4.85 5.78 19.0% AvgSalesRank 84674 27450 67.6% Music

AvgRecommendedProducts 2.43 13.96 5.7 X AvgCategoricalSimilarity 4.92 5.67 15.2% AvgSalesRank 26739 8217 69.3% Video

AvgRecommendedProducts 1.95 12.73 6.5 X AvgCategoricalSimilarity 3.56 5.83 63.7% AvgSalesRank 7196 2821 60.8% DVD

AvgRecommendedProducts 3.43 14.99 4.4 X AvgCategoricalSimilarity 5.03 6.08 20.9% AvgSalesRank 8430 2646 68.6% SincetheexamplewehaveusedisonBooksletmeexplorethebookscategoryinmoredetailtoexplainthistable.Whenwestarted,onaveragetherewere2.75recommendationsforeachbook,notnearenoughtoprovideadequatechoices,afterthetransformation,wenowhave14recommendationsperbook,thatismuchmoreusefulforaneCommerceengine.Thisalsoensuresalargersetofchoicesprovidedtoconsumers.AsameasureofrelevancewehaveusedCategoricalSimilarityandonthismeasurethereisa19%improvementinCategoricalSimilarity.TheSalesRankforaproductiscomputedbyAmazonasafunctionofhistoricalsales.AlowerSalesRanksignifiesahighersellingproduct.ThereisaniceimprovementintheaverageSalesRankby67%increasingtheprobabilityofsellingotherproducts.Nowletmechallengesomeoftheassumptionsmadeinthisproject:

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- increaseaverage#ofrecommendedproductsto~15,isthisareasonablenumber,doweneedmore,doweneedless;inthiscasewiththesamealgorithmthiscanbechanged

- the“weight”ofeachLevelinthecategorytreeisthesame,wecouldweighdeepersub-categorieshigherandseeifthatyieldsbetterresults,thisisworthexperimentingwith

- loweringtheaveragePageRankoftheco-sellingproductscouldleadtoarichgetricherphenomenon,isthiswhatwewantordowewanttosprinkletherecommendationswithsomehighsellersandsomelowsellers,thisiscertainlyastrategythatcanbeinvestigated

- isCategoricalSimilarityagoodmeasure,theresultsseempromising,butneedstobetested- tothelastpointandeverythingstatedsofartheultimatemeasureiswillthisleadtomore

ecommercesalesandtheonlywaywedeterminethatisbyrealworldtestsofthemodel- finallytheassumptionmadeinthisdiscussionaboutmaximizingsalesistomaximizethe

numberofitemssold,wouldwedoanythingdifferenttomaximizesalesrevenues,wesuspecttheapproachissimilar,butthiswouldbeanotherworthypathtoinvestigate

5.0ConclusionWestartedwithaDatasetthatprovidedasparsesetofrecommendationsforco-sellingproducts.Thisisarealworldscenariowhereoneneedstoproducemoreandrelevantrecommendationsgivenalimitedsetofdata.Welookedatclustermodelsandcollaborativefiltering,bothofwhichwouldrequiremoreinformationoncustomersthatwasnotavailable.Sowiththelimitedinformationavailable,wecameupwithametricaroundCategoricalSimilarityandprovidedadditionalrecommendationsthatimproveduponboththismeasureandtheaverageSalesRank.Nowinarealworldscenariothenextstepistorunatestforalimitedperiodoftime,sowecanprogressivelyrefineitbasedonrealresults.Thatsaidourtheoreticalresultsshowthatweareheadedintherightdirection.6.0References

1. J.Breese,D.Heckerman,andC.Kadie,“EmpiricalAnalysisofPredictiveAlgorithmsforCollaborativeFiltering,”Proc.14thConf.UncertaintyinArtificialIntelligence,MorganKaufmann,1998,pp.43-52.

2. B.M.Sarwaretal.,“Item-BasedCollaborativeFilteringRecommendationAlgorithms,”10thInt’lWorldWideWebConference,ACMPress,2001,pp.285-295.

3. M.BalabanovicandY.Shoham,“Content-BasedCollaborativeRecommendation,”Comm.ACM,Mar.1997,pp.66-72.

4. L.UngarandD.Foster,“ClusteringMethodsforCollaborativeFiltering,”Proc.WorkshoponRecommendationSystems,AAAIPress,1998.

5. LindenG.,SmithS.&YorkJ,–Amazon.comRecommendations:ItemtoItemCollaborativeFiltering,2003,