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Algebraic-Geometric Methods for Complexity Lower Bounds

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Algebraic-Geometric Methods for Complexity Lower Bounds
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DANDEKAR, PRANAV ( Author, Primary )
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2004

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ALGEBRAIC-GEOMETRICMETHODSFORCOMPLEXITYLOWERBOUNDS By PRANAVDANDEKAR ATHESISPRESENTEDTOTHEGRADUATESCHOOL OFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENT OFTHEREQUIREMENTSFORTHEDEGREEOF MASTEROFSCIENCE UNIVERSITYOFFLORIDA 2004

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Copyright2004 by PranavDandekar

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Idedicatethisworktomyparents.

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ACKNOWLEDGMENTS Iwouldliketoexpressmydeepestgratitudetomyadvisor,Pr ofessorMeeraSitharam. Thoughitisaclichetosayso,thisworkwouldhavebeenimpo ssiblewithouthervery activeinvolvement,constantencouragementandtheinspir ationIdrewfromher.Thebest analogyforwhatshetaughtmeissomeoneteachingatoddlert orstcrawl,thenstand,walk andrunbyholdinghishand.Workingwithherhasbeenawonder fullearningexperience. Sheiseverythingagreatadvisoris,andthensome. IamthankfultoProf.TimDavisandProf.AndrewVinceforser vingonmysupervisorycommittee,andforposingsomethought-provokingques tionsaboutmywork.Iwould alsoliketothankmyfriendshereatUF,speciallySrijitKam athandAndrewLomonosov, forhelpingmewritethisthesisandfeigninginterestinmyw orkduringidlediscussions. iv

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TABLEOFCONTENTS page ACKNOWLEDGMENTS .................................iv ABSTRACT ........................................vii CHAPTER1INTRODUCTION ..................................1 2 MATHEMATICALPRELIMINARIES .....................3 2.1Matrices ....................................3 2.2InnerProductsandNorms ..........................4 2.3GeometricConcepts .............................5 3 APPLICATIONINCOMPUTERSCIENCE .................7 3.1CommunicationComplexity .........................7 3.2ThresholdCircuitComplexity ........................8 3.3MaximalMarginClassiers .........................8 3.4GeometricEmbeddings ............................9 4 REALIZATIONSANDREALIZABILITY ..................12 4.1HalfSpaceRealization ............................12 4.1.1KnownBounds ............................12 4.1.2GeneralizedLowerBound ......................14 4.2ANewNotionofRealization ........................15 4.2.1ConstructingBfromForster'sRealization .............15 4.2.2NicenessPropertiesofRealizationSubspaces ............16 4.2.3RephrasingForsterinTermsof B ..................17 4.2.4RephrasingtheGeneralizedLowerBoundinTermsof B .....20 4.2.5IntuitiveJustication .........................21 5 NICEREALIZATIONS ..............................22 5.1NewDenitionofNiceness ..........................22 5.1.1ImprovedBoundsfor M 2f 1 ; +1 g n m ..............22 5.1.2ImprovedBoundsfor M 2 R n m ..................25 5.1.3WhyIsThisBoundBetter? .....................28 5.2NicePropertiesofSpecicConstructions ..................28 5.2.1Forster'sConstruction ........................28 5.2.2Forster'sconstructionhasproperty5.1 ...............29 5.2.3Belcher-Hicks-SitharamConstruction ................31 5.2.4Belcher-Hicks-Sitharamconstructionhasproperty5 .1 ......31 v

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6 THETOPOLOGYOFBALLS .........................34 6.1TheGeneralizedAlmost-SphericalProperty ................34 6.2ObservationsAboutTheGeneralizedAlmost-sphericalP roperty .....35 6.3ObservationsontheFigure .........................39 7 CONCLUSIONANDFUTUREWORK ....................42 REFERENCES .......................................45 BIOGRAPHICALSKETCH ...............................46 vi

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AbstractofThesisPresentedtotheGraduateSchool oftheUniversityofFloridainPartialFulllmentofthe RequirementsfortheDegreeofMasterofScience ALGEBRAIC-GEOMETRICMETHODSFORCOMPLEXITYLOWERBOUNDS By PranavDandekar December2004 Chair:MeeraSitharamMajorDepartment:ComputerandInformationScienceandEng ineering Complexitylowerboundsisthestudyofthe intrinsiccomplexity ofcomputational problems.Lowerboundargumentsprovethe(in)solvability ofaprobleminacertain time/spaceassumingacertainmodelofcomputationfor all validprobleminstances.In thisworkwestudyageometriclowerboundproblem{tondthe minimaldimensionof asubspacethatintersectsagivensetofcells(orthants)in R m .Thisproblemunderlies manyproblemsindiverseareasofComputerSciencesuchasCo mmunicationComplexity, LearningTheory,GeometricEmbeddings,etc.Westartbyrep hrasingtheproofofalongstandingopenprobleminCommunicationComplexity.Rephra singtheproblemintermsof realizationsubspaces leadsustosomeinterestingpropertiesofthesesubspaces. Werelate thesepropertiesandusethemtoimprovetheknownresult.We alsoposesomequestions aboutthesesubspacesandtheirrelationtosomeresultsinf unctionalanalysis.Finally,we identifythevariousdirectionstoproceedinusingthisgeo metriclowerboundproblemasa startingpoint. vii

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CHAPTER1 INTRODUCTION ComplexitylowerboundsistheareaoftheoreticalComputer Sciencethatstudiesthe intrinsiccomplexity ofcomputationalproblems.Inotherwords,lowerboundprob lems requireustoshowthatacertaincomputationalproblemneed s atleast acertainamountof resourcesassumingacertainmodelofcomputation.Lowerbo undargumentsareingeneral hardtomakebecauseeachsuchargumentisasweepingstateme ntaboutthe(in)solvability oftheprobleminacertaintime/spacefor all validprobleminstances.Forexample,there areanumberofwell-knownsortingmethodssuchasBubbleSor t,QuickSortetc.butthe argumentthatnocomparison-basedsortingmethodcantakel essthan O ( n log n )timeona random-accessmachineisdecidedlyhardertomake. Lowerboundproblemsariseinvariouscontexts-therearean umberoflowerbound questionsoncircuitsize/depth,manysuchquestionsongra phs(mostofwhichareknown tobe NPcomplete )andmanyinvolvingmatrices.Inparticular,severalmatri xfunctions relatedtorankanditsrobustnessundervarioustransforma tionsofthematrixhavebeen extensivelyusedinmanycomplexityproblems.Matrixrelat edproblemslendthemselves totheuseofeleganttechniquesfromalgebraic-geometry. Thisworkstudiesanumberofmatrixrelatedlowerboundprob lemsarisingindiverse areas.Further,itestablishestheinterconnectionsbetwe enthemandusestechniquesfrom algebraicgeometrytoexplaintheresultsmoreintuitively andtoimprovetheknownresults. Thebasicgeometricproblemweseektoanswercanbestatedsi mplyas: Givenasetof n orthants(anorthantisageneralizationofaquadrantin2Dt o m dimensions)in R m , whatisthelowestdimensionalhomogeneoussubspaceof R m thatintersectstheseorthants. Statedinlinearalgebraicterms,weask: Givenamatrix M 2f 1 ; +1 g n m ,whatisthe lowestrankmatrix M 0 2 R n m suchthatsign ( M 0 ij )= M ij forall i;j . Ononehandthis isthecommonproblemunderlyingmanyproblemsindiversear easofComputerScience, 1

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2 andontheotherhanditallowstheuseofalgebraic-geometri cconceptssuchas Grassmann manifolds andthetopologyof L 1 ;L 2 and L 1 balls. Thethesisisorganizedasfollows:Wemotivatethestudyoft hisgeometricproblemin Chapter3bymentioningitslinkswithlowerboundquestions inareassuchasCommunicationComplexity,ThresholdCircuitComplexity,LearningT heoryetc.Wethenpresentthe originallongstandingopenprobleminCommunicationCompl exityanditsrecentsolution byJurgenForster[ 1 ]inChapter4.Forsterdenestheproblemasthatofndingal owdimension hyperplanearrangement thatrepresents M andcallsitageometric realization of M .Forsterusesthenotionofa`nice'realization-realizati onthathasanon-trivial propertythathelpsattaingoodbounds.WerephraseForster 'sresultanditsproofinmore intuitivetermsandprovideajusticationforit.Wealsode neafewnicepropertiesofour own,oneofwhichisarelaxationofForster'spropertyandth eotherhelpsusarriveata newresult.Thestudyoftheseniceproperties,theirrelati onships,showingtheirexistence andanalysingwhattheybuyushasbeenthecentralthreadand themajorcontribution ofthiswork.WedeneandrelatethesepropertiesinChapter s4and5andshowanew boundontheminimaldimensioninChapter5.Wealsogiveanin tuitiveexplanationof whatourresultmeansandwhyitisbetter.Weposeaquestiona boutthecontainment ofthe L 1 ballin L 2 ballthatstemsfromoneofthe`nice'propertiesandtrytoan swerit inChapter6.Finally,weidentifythedierentdirectionsi nwhichonecoulddelvedeeper usingthisproblemasastartingpointinChapter7.

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CHAPTER2 MATHEMATICALPRELIMINARIES Inthischapterwexsomenotationfortherestofthethesis, andquotesomerelevant resultsfromlinearalgebraandfunctionalanalysisthatwe willuselater. 2.1Matrices R n m isthesetofall n m matricesdenedoverthereals.Ingeneralif F denotesa eld,then F n m denotesthesetofall n m matricesover F .Inthiswork,weusuallytalk aboutmatricesdenedoverthereals( M 2 R n m )or 1matrices( M 2f 1 ; 1 g n m ).The entryinthe i throwand j thcolumnof M isdenotedby M ij .Weuse i toindextherows and j toindexthecolumnsofthematrix. Therankofamatrix M isthenumberoflinearlyindependentrowsorcolumnsin M . Theproductoftwomatrices A 2 R n m and B 2 R m p isthematrix C = A:B 2 R n p where C ik = P j A ij B jk ;i 2f 1 ::n g ;j 2f 1 ::m g ;k 2f 1 ::p g . TheKroneckerproduct(ortensorproduct)oftwomatrices A 2 R n m and B 2 R p q isdenedas: A O B = 2666666666666664 A 1 ; 1 BA 1 ; 2 B:::A 1 ;n B A 2 ; 1 BA 2 ; 2 B:::A 2 ;n B ::: A m; 1 BA m; 2 B:::A m;n B 3777777777777775 2 R np mq suchthatrank( A N B )=rank( A ).rank( B ). A squarematrix isa m m matrix.The identitymatrix I isasquarematrixsuch that I ij =1if i = j and0otherwise.Theinverseofasquarematrix A isthematrix A 1 suchthat AA 1 = I 3

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4 Asquarematrix A hasaninverse i j A j6 =0.Amatrixthathasaninverseiscalled non-singular orinvertible.Thegroupofnon-singularmatrices A 2 R m m isdenotedby GL ( m ).Thetraceofasquarematrix A 2 R m m isdenedas trace( A )= m X i =1 A ii For A 2 R m m ,ifthereisavector x 2 R m 6 =0suchthat A x = x forsomescalar ,then iscalledthe eigenvalue of A withthecorresponding eigenvector x . Thetransposeofamatrix A 2 R n m isamatrix A | 2 R m n suchthat A ij = A |ji .In particular,thetransposeofacolumnvectorisarowvector. Amatrix A iscalled orthogonal if AA | = I .Asquarematrix A 2 R m m iscalled symmetric if A = A | .Asymmetric matrixissaidtobe positivesemi-denite ,denotedby A 0,ifallofitseigenvaluesare non-negativeorequivalently,if x | A x 0forall x 2 R m .Asymmetricmatrixissaidtobe positivedenite ifallofitseigenvaluesarepositive. Aspecialkindofmatrixthatwewillencounteroftenisthe Hadamard matrix.Hadamard matrices H n 2 R 2 n 2 n areexamplesofmatriceswithpairwiseorthogonalrowsandp airwise orthogonalcolumns.TheyaredenedintheSylvesterformas H 0 =1 ;H n +1 = 0B@ H n H n H n H n 1CA : 2.2InnerProductsandNorms The innerproduct oftwovectors v ; w 2 R m isdenedas h v ; w i = m X i =1 v i w i Thevectors v , w arecalled orthogonal if h v ; w i =0.Foravector x 2 R m ,thevector norm L p isdenedas jj x jj p = n X i =1 j x i j p ! 1 =p

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5 for p =1 ; 2 ; 3 ;::: .Themostcommonlyencounterednormsare: 1. L 1 (or Manhattan )norm: jj x jj 1 = P i j x i j .Itisused,forexample,incomputingthe Hammingdistancebetweentwobitstrings. 2. L 2 (or Euclidean )norm: jj x jj 2 = q P i x 2i .Itisthemostcommonmeasureofdistance betweentwopoints. 3. L 1 norm: jj x jj 1 =max i j x i j . Thevectors v , w arecalled orthonormal if h v ; w i =0and jj v jj 2 = jj w jj 2 =1. Theoperatornormofamatrix A 2 R n m is jj A jj =sup x 2 R m jj A x jj 2 jj x jj 2 Foramatrix A 2f 1 ; 1 g n m ,itiseasytoseethat jj A jj = p m iftherowsof A are pairwiseorthogonaland jj A jj = p n ifthecolumnsof A arepairwiseorthogonal.Itfollows that jj H n jj =2 n 2 .Alsoifrank( A )=1then jj A jj = p mn . Itiswellknownthatforanymatrix A 2 R n m , jj A jj 2 = jj A | A jj = jj AA | jj .Forevery symmetricmatrix A 2 R m m thereisanorthonormalbasis d 1 ;d 2 ;:::;d m of R m andthere arescalars 1 ; 2 ;:::; m 2 R m (theeigenvaluesof A )suchthat A = m X i =1 i d i d |i Thesignumfunctionsign: R ! R ,whichweencounteroften,isdenedas sign( x )= 8>>>><>>>>: 1 ;x> 0 ; 0 ;x =0 ; 1 ;x< 0 2.3GeometricConcepts R m isthe m -dimensionalrealspace-thesetofall m -dimensionalvectorsdenedover thereals.Weuse m and k forthedimensionofaspace.Thusamatrix M 2 R n m canbe treatedasasubspaceof R m ,i.e.asetof n vectorsin R m .Weusethematrixandsubspace notioninterchangeably. Therankofamatrix M isthesameasthedimensionofthesubspacedenotedby M .Abasisofavectorspace V isdenedasasubsetofvectorsin V thatarelinearly

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6 independentandspan V ,i.e.everyvectorin V canbewrittenasalinearcombinationof thebasisvectors.Thenumberofbasisvectorsisequaltothe dimensionof V . A hyperplane isageneralisationofanormaltwo-dimensionalplaneinthr ee-dimensional spacetoits( m 1)-dimensionalanaloguein m -dimensionalspace.Eachhyperplaneseparatestheambientspaceintotwo half-spaces .Thetermshyperplaneandhalf-spaceareused interchangeablyintheliterature.Ahyperplane(orhalf-s pace)iscalledhomogeneousifits boundarycontainstheorigin. An orthant isthesetofallvectorsthathavethesamesigns.Forexample ,in2Dwe have4orthants(commonlycalledquadrants)andin3Dwehave 8orthants(commonly calledoctants).Ingeneral,thereare2 m orthantsin m dimensions,eachofwhichcanbe uniquelyidentiedbya m -longsignvector. The( m 1)-dimensionalsphere( unitball in m dimensions)istheset f x 2 R m : jj x jj 2 = 1 g andisdenotedby S m 1 . Ingeneral, L p = d ballin R m isdenedastheset f x 2 R m : jj x jj p = d g .Wealsouse L mp tomeanthespace R m under L p norm. An arrangementofEuclideanhalf-spaces that represents M 2 R n m consistsofpoints u i 2 R k foreveryrow i 2f 1 ::n g of M andofhalf-spaces H j R k foreverycolumn j 2f 1 ::m g of M suchthat u i liesinthehalf-space H j ifandonlyifsign( M ij )=+1. k is calledthe dimension ofthearrangment. Ahomogeneoushalf-space H isoftheform H = f z 2 R k : h z ; v i 0 g forsomevector v normal totheboundaryofthehalf-space.Thenwedeneanarrangeme nt ofhomogeneousEuclideanhalf-spacesas:Denition. AnarrangementofhomogeneousEuclideanhalf-spacesrepre sentingamatrix M 2 R n m withnozeroentriesisgivenbythecollectionofvectors u i ; v j 2 R k for i 2 f 1 ::n g ;j 2f 1 ::m g suchthatsign( M ij )=sign h u i ; v j i . Hereeachvector v j isnormaltoanduniquelydenesthehomogeneoushalf-space H j .

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CHAPTER3 APPLICATIONINCOMPUTERSCIENCE Inthischapterwemotivatethestudyofthegeometricproble mstatedintheintroduction-namely,determiningtheminimaldimensionofthesubs pacethatintersectsagivenset oforthants-bystatingitsapplicationsindiverseareasof ComputerScience.Thischapter shouldalsoservetoholdtheinterestofthereaderinthiswo rkwhovaluesaproblemby itsusefulnessintherealworld,andtojustifythestudyoft hisproblemtothosewhodon't nditsmathematicaleleganceadequatelyappealing. 3.1CommunicationComplexity Considerthetwo-partycommunicationcomplexityproblemtwoparties A and B want tocompute f ( x;y ),where f isadistributedfunction f : f 0 ; 1 g n f 0 ; 1 g n !f 0 ; 1 g .Both haveunboundedcomputationalpower. A hasinputstring x andBhasinputstring y .A two-wayprobabilisticcommunicationprotocolisaprobabi listicalgorithmusedby A and B tocompute f byexchangingmessages. A and B taketurnsprobabilisticallychoosing amessagetosendtotheotherpartyaccordingtotheprotocol .Wesaythataprotocol computesthedistributedfunction f : f 0 ; 1 g n f 0 ; 1 g n !f 0 ; 1 g withunboundederroriffor allinputs( x;y ) 2f 0 ; 1 g n f 0 ; 1 g n thecorrectoutputiscalculatedwithaprobabilitygreater than 1 2 .Thecomplexityofacommunicationprotocolis d log 2 N e ,where N isthenumber ofdistinctmessagesequencesthatcanoccurincomputation sthatfollowtheprotocol.The communicationcomplexity ~ C f ofadistributedfunction f : f 0 ; 1 g n f 0 ; 1 g n !f 0 ; 1 g isthe smallestcomplexitythatacommunicationprotocolfor f canhave. Toestablishtheconnectionbetweentheproblemofestimati ngtheminimaldimension of 1matricesandthecommunicationcomplexityproblem,wenot ethatafunction f : f 0 ; 1 g n f 0 ; 1 g n !f 0 ; 1 g inducesamatrix M 2f 1 ; 1 g 2 n 2 n suchthat M ij =2 f ( x;y ) 1 where x and y arethebinaryrepresentationsofintegers i and j respectively.Paturiand Simon[ 2 ]showedthattheminimaldimension d ( f )ofthematrixinducedbythefunction 7

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8 f isrelatedtoitscommunicationcomplexity ~ C f as: d log d ( f ) e ~ C f d log d ( f ) e +1 Alon,FranklandRodl[ 3 ]showedthatfor almostall n n matriceswith 1entries,the minimaldimensionisatleastn( n )implyingthatformostBooleanfunctionstheunbounded errorprobabilisticcommunicationcomplexityisasymptot icallyaslargeasitcanbe(that is,linearinthelengthoftheinput).However,provingeven asuperlogarithmiclowerbound ontheminimaldimensionofanexplicitmatrixremainedadi cultopenquestion.Jurgen Forster[ 1 ]solvedthislongstandingopenquestionbyshowingagenera llowerboundonthe minimaldimensionofamatrixintermsofitsoperatornorm.A sacorollary,hederiveda lowerboundof p n ontheminimaldimensionofan n n Hadamardmatrix.Later,Forster, Krause etal [ 4 ]generalizedForster'sboundtorealmatrices(hence,tofu nctionsdened overthereals)withnon-zeroentries. Oneofthemotivationsbehindstudyingthisproblemwastore placethedependency ontheoperatornormof M bysomeotherpropertythatcapturestheorthantstructure representedbytherowsof M ,becauseintuitivelytheminimaldimensionshoulddependo n thesetoforthantsrepresentedby M . 3.2ThresholdCircuitComplexity Alinearthresholdgateisabooleangatethatoutputsa1ifth eweightedsumof itsinputsisgreaterthanathreshold t and0otherwise.Forster,Krause etal [ 4 ]proveda lowerboundonthesizeofdepth-2thresholdcircuitswheret hetopgateisalinearthreshold gatewithunrestrictedweightsandthebottomlevelgatesar elinearthresholdgateswith restrictedintegerweights.Theresultimpliesthat(loose lyspeaking)booleanfunctionswith `high'minimaldimensioncannotbecomputedby`small'dept h-2circuitshavingtheabove restrictions.MeeraSitharam(personalcommunication)ob servedthattherestrictionon theweightsofthebottomlevelgatesisnotrequiredtoprove theresult. 3.3MaximalMarginClassiers Computationallearningtheoryisthestudyofecientalgor ithmsthatimprovetheir performancebasedonpastexperience.Aclassofproblemsin learningtheoryisclasscation problems.Thelearningalgorithmis`trained'onsometrain ingdatawhichisasetofdata

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9 instancesandtheirclassesorlabels.Afterthealgorithmi strained,itisthenpresented withunlabeleddataandisrequiredtolabel(orclassify)it accurately.Themostimportant testoftheusefulnessofalearningalgorithmisitsperform anceinexperimentsonreal-world data.Recentlytherehasbeenalotofinterestinmaximalmar ginclassierswhichhave shownexcellentempiricalperformance.Maximalmargincla ssiersarelearningalgorithms thatcomputethehyperplanethatseparatesasamplewiththe largestmarginandusethis hyperplanetoclassifynewinstances.Oftenthesetofinsta ncesismappedtosomepossibly highdimensionbeforethehyperplanewiththemaximalmargi niscomputed.Ifthenorms oftheinstancesareboundedandahyperplanewithalargemar gincanbefound,then thatgivesaboundontheVapnik-Chervonenkis(VC)dimensio noftheclasser.TheVCdimensionisameasureofthepowerofthelearningalgorithm ;asmallVC-dimensionmeans thataclasscanbelearnedwithasmallsamplesize.Knowingo nlytheVC-dimensionand thetrainingerror,itispossibletoestimatetheerroronth efuturedata. Thesuccessofmaximalmarginclassiersraisesthequestio nwhichconceptclasses canberepresentedbyanarrangementofEuclideanhalfspace swithalargemargin.We saythatthematrix M 2 R n m withnozeroentriescanberealizedbyanarrangement ofhomogeneoushalfspaceswithmargin r iftherearevectors u i ; v j 2 S k 1 (where k canbearbitrarilylarge)suchthatsign( M ij )=sign h u i ; v j i and jh u i ; v j ij r forall i;j . Interpreting v j asthenormaltotheboundaryofthe j thhalfspace,sign( M ij )=sign h u i ; v j i meansthatthevector u i liesinthehalfspacecontaining v j ifandonlyif M ij ispositive. Therequirement jh u i ; v j ij r meansthatthepoint u i hasdistanceatleast r fromthe boundaryofthehalfspace. Forster[ 1 ]showsthatthelowerboundontheminimaldimension d ( M )istheinverse oftheupperboundonthemaximalmargin r .Therefore,atighterlowerboundonthe minimaldimensionwilldirectlyresultinatighterupperbo undonthemaximalmargin. 3.4GeometricEmbeddings Geometricembeddingslieattheintersectionofanalysisan dgeometryandhaveattractedmuchattentioninrecenttimesduetotheiralgorith micapplicationsindatamining andinformationextractionfrommassivemultidimensional datasets.Specically,thegeometricembeddingsthatleadto dimensionalityreduction (whichwewilldeneshortly)are

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10 theonesofgreatinterestforecientlysolvingproblemsin datamining,proteinmatching etc. A metricspaceM (alsocalleda metric )isapair( X;D ),where X isasetof points and D : X X ! [0 ; 1 )isa distancefunction satisfyingthefollowingpropertiesfor p;q;r 2 X : D ( p;q )=0i p = q D ( p;q )= D ( q;p ) D ( p;q )+ D ( q;r ) D ( p;r ) Formally,geometricembeddingsaremappings f : P A ! P B suchthat P A isasetofpointsintheoriginalspace(whichisusuallyamet ricspaceandtherefore hasadistancefunction D ( ; )) P B isasetofpointsinthehostspace(whichisusuallyanormeds pace L ds ) forany p;q 2 P A wehave 1 =c D ( p;q ) jj f ( p ) f ( q ) jj s D ( p;q ) foracertainparameter c calleddistortion. Anyembedding f : A ! B canbeclassiedbasedonthetypesofspaces A and B . B is usuallyanormedspace L d 0 p 0 . A canbeanyofthefollowing: 1. A isanitemetric M =( X;D )inducedbygraphs,e.g.obtainedbycomputing all-pairsshortestpaths.Themainapplicationsofsuchemb eddingsareapproximationalgorithmsforoptimizationproblemsongraphs.Other applicationsinclude proximity-preservinglabelingandprovinghardnessofapp roximation. 2. A isasubsetof L dp .Heretheintentistomapthe whole normedspaceintothehost spaceasopposedtoanitesetofpoints.Thisisusefulincas eswhenallthepoints inthesetarenotknowninadvance,e.g.whenthepointsconst ituteasolutionto anNP-hardproblem,orwhentheyaregivenonlinebytheuser. Herewehavetwo scenarios: (a) d 0 <
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11 preservingimportantcharacteristicsofthepointset.Thi sleadstoaspeedupof algorithmswhoserunningtimedependsonthedimension. (b) p 6 = p 0 (andmostoften d 0 >>d ):Suchembeddingsallowustoswitchfrom \dicult"norms(e.g. L 2 )to"easier"norms(e.g. L 1 ). 3. A isaspecialmetric,usuallymoregeneralthananorm.Exampl esofsuchmetrics arethe editmetric (deningsimilaritybetweenstringsofcharacters), Hausdormetric (deningsimilaritybetweensetsofpoints)and Hammingmetric (deningthe Hammingdistancebetweenasetofpointsover F 2 ). Forsterformulatedtheproblemofestimatingtheminimaldi mension k ofamatrix M 2 R n m asaproblemto realize thematrixbyalineararrangementofhomogeneoushyperplanesin k dimensions.Nowtreatingeachrowof M asavectorin R m ,wenoticethatthe hyperplanearrangementinducesaHammingmetricontheseto f n vectors(eachvector M i ;i 2f 1 ::n g correspondstoapoint p i 2f 1 ; 1 g m suchthat p i =(sign h M i ; v j i : j 2 f 1 ::m g )where v j isthenormaltothe j thhyperplane). Thusouroriginalproblemcannowbecastasaproblemofembed dingan m -dimensional Hammingmetricinducedbythelinearhyperplanearrangemen tintoaHammingmetricin k dimensionsinducedbythesamehyperplanearrangementwith outanydistortion.This problemhasdirectapplicationsinproteinmatching[ 5 ].

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CHAPTER4 REALIZATIONSANDREALIZABILITY Inthischapter,wediscusstheknownresultsontheproblemo utlinedintheintroductionandrephrasesomeoftheminmoreintuitivetermsusi ngtechniquesthatleadto improvedbounds.Westartbydeningtheconceptofrealizat ionofamatrix M ,rstin termsofhalf-spacesandlaterredeningitintermsofsubsp acesthatintersectthesameset oforthantsasdenedbytherowsof M . 4.1HalfSpaceRealization Thelongstandingopenquestionontheminimaldimensionofm atriceswasrstposed byPaturiandSimon[ 2 ].Forster[ 1 ]whosolvedtheproblembyshowingalowerbound ontheminimaldimensionof M intermsoftheoperatornormof M denestheproblem asrealizing M intermsofalineararrangementofhomogeneoushalfspaces. Werestate Forster'sdenitionofrealization:Denition. Amatrix M 2 R n m withnozeroentriescanbe realized bya k -dimensional lineararrangementofhomogeneoushalfspacesiftherearev ectors u i ; v j 2 R k for i 2 f 1 ::n g ;j 2f 1 ::m g suchthatsign( M ij )=sign h u i ; v j i . Thevector v j isinterpretedasanormaltotheboundaryofthehomogenoush alf-spaceand henceuniquelydenesit.Thevector u i denesapointthatliesinthehalfspacedened by v j ifsign h u i ; v j i =+1andintheoppositehalfspaceifsign h u i ; v j i = 1. 4.1.1KnownBounds Thefollowinglowerboundontheminimaldimensionwasprove drecentlybyForster [ 1 ]: Theorem4.1(Forster[ 1 ]) Ifamatrix M 2f 1 ; 1 g n m canberealizedbyanarrangementofhomogeneoushalfspacesin R k ,then k p mn jj M jj (4.1) 12

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13 Proofidea. Forsterusesabridgingquantitytoprovehisresult.Heshow sanupperbound onthebridgingquantitythatholdsforanyrealizationofth egivenmatrix M .Healsoshows alowerboundonthebridgingquantitythatholdsforonlythe `nice'realizationsof M .He thenshowsthatgivenanyrealization,hecaniterativelyn donethatisniceandtherefore satisesthelowerboundonthebridgingquantity.Whilerep hrasingForster'sproof,we alsodeneafewnicepropertiesofourrealizations,namely ,thealmost-sphericalproperty whichweshowtobearelaxationofForster'spropertyandthe projection-typeniceproperty whichhelpsuschangeForster'supperboundonthebridgingq uantity.Thestudyofthese nicepropertiesofrealizations,relatingthem,showingth eirexistenceandndingoutifthey buyusbetterresultsisthecentralthemeofthiswork.Proof(byForster). Assumethattherearevectors u i ; v j 2 R k suchthatsign h u i ; v j i = M ij .Forsterobservedthatnormalizing u i and v j willnotaectthesignoftheinnerproduct h u i ; v j i .Thereforewemayassumethat u i ; v j 2 S k 1 .Healsoobservedthatforany nonsingularlineartransformation A 2 GL( k ),replacing u i ; v j by( A | ) 1 u i ;A v j alsodoes notaectthesignoftheinnerproduct.Hethenshowsthatthe reexistsalinearmapping thattransformsagivensetof v j 'sintoasetof\nice" v j 'ssuchthat P j v j v j | = m k I k ,i.e. thevectors v j arenicelybalancedinthissense.Forsuchnice v j 'sitholdsthat, X j jh u i ; v j ij X j h u i ; v j i 2 = u i | 0@ X j v j v j | 1A u i = m k (4.2) Inequality( 4.2 )meansthatforall j ,theabsolutevaluesoftheinnerproducts h u i ; v j i areontheaverageatleast 1 k ,i.e.thevectors u i cannotliearbitrarilyclosetothehomogeneoushyperplanewithnormal v j . Forstergivesacorrespondingupperboundintermsoftheope ratornorm jj M jj ofthematrix ontheabsolutevaluesofthescalarproducts h u i ; v j i as: X i 0@ X j jh u i ; v j ij 1A 2 m jj M jj 2 (4.3)

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14 From( 4.2 )and( 4.3 ),itfollowsthat n m k 2 X i 0@ X j jh u i ; v j ij 1A 2 m jj M jj 2 (4.4) Thelowerbounddirectlyfollows. 4.1.2GeneralizedLowerBound Forster,Krause,Lokam etal [ 4 ]generalizedthelowerboundresultobtainedin[ 1 ] torealmatrices.ThisresultwidenedtheapplicabilityofF orster'sboundfromboolean functionstofunctionsdenedoverthereals.Alsothegener alizedresultgivesaccurate boundsfororthogonalmatricesthathaveasmallnormwhicht hepreviousresultfailsto achieve.Thegeneralizedlowerboundresultcanbestatedas : Theorem4.2 Let M 2 R n m beamatrixwithnozeroentries.Thenthelowestdimension k inwhich M canberealizedbyalineararrangementofhomogeneoushalfs pacesisgiven by: k p mn jj M jj min i;j j M ij j (4.5) Proof TheproofisalongthelinesofForster'soriginalresult.It assumesthatthereare vectors u i ; v j 2 R k suchthatsign( M ij )=sign h u i ; v j i forall i;j .UsingForster'sargument, itisassumedthatagivenlineararrangement u i ; v j canbenormalizedsuchthat u i ; v j 2 S k 1 and X j v j v j | = m k I k Letmin i;j j M ij j = M min .Thenforall j ,itholdsthat X j M ij h u i ; v j i = M min X j M ij M min h u i ; v j i M min X j h u i ; v j i 2 = M min : u i | 0@ X j v j v j | 1A u i or, X j M ij h u i ; v j i M min m k (4.6) [ 4 ]alsoprovesthesameupperboundonthenewbridgingquantit y.Specicallythat X i 0@ X j M ij h u i ; v j i 1A 2 m jj M jj 2 (4.7)

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15 Itistobenotedthattheupperbounddoesnotdependuponthem inimumentryin M . From( 4.6 )and( 4.7 ),weget M 2 min n m k 2 X i 0@ X j M ij h u i ; v j i 1A 2 m jj M jj 2 (4.8) Thegeneralizedlowerboundresultfollows. 4.2ANewNotionofRealization Wecharacterizearealizationofthematrix M 2 R n m intermsofthe k -dimensional subspaceof R m thatcutsthroughthesetoforthantsdenedbythesignsofth erows in M .Themotivationbehindthisnewnotionofrealizationistoc astForster'sresultin moresimpliedtermsasananswertothequestionsposedbyus intheintroduction.Also, thisnotionofrealizationleadsustosomeinterestingprop ertiesofsubspaces,andtoan improvedresult.Denition. Amatrix M 2 R n m withnozeroentriescanbe realized bya k -dimensional subspace B ifthereexistvectors w i = a i B ,where a i 2 R k ,suchthatsign( w i )=sign( M i ) forall i . 4.2.1ConstructingBfromForster'sRealization ToestablishanequivalencebetweenForster'sdenitionof realizationandournotionof realization,werstgiveaconstructiontoobtainthe B matrixandthe w i 'sfromForster's u i and v j vectors. Lemma4.3 Given M 2 R n m and u i ; v j 2 S k 1 suchthatsign h u i ; v j i =sign ( M ij ) for all i;j ,constructthe k m matrix B suchthatthe v j 'sarethecolumnsof B andlet w i = p k=m u i B .Thentherowsofthematrix B formabasisforthe k -dimensional subspacethatrealizes M . Proof Wenotethatvectors w i = u i B areactuallythevectors u i expressedinthebasis B . Inotherwords,vectors w i arenothingbut u i livingintheambient m -dimensionalspace. Thescalingof w i byafactorof p k=m isneededlaterinrephrasingForster'sproofwherethe w i 'sarerequiredtobeunitvectors.Now,the j thco-ordinateof w i , w i ( j )= p k=m h u i ;B j i . Sincethecolumns B j ofthematrixareactuallythevectors v j byconstruction,therefore

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16 wehavethat sign( w i ( j ))=sign r k m h u i ;B j i ! =sign r k m h u i ; v j i ! =sign( M ij ) forall i;j .Hencethis B satisesthedenitionofasubspacerealizingthematrix M . Inthesequel,wewilltalkaboutrealizationsintermsofvec tors w i andthesubspace B ratherthanvectors u i and v j . InordertorephraseForster'sproofofthelowerboundresul t,werstdenefew nicenesspropertiesofourrealizationsubspace B andshowtheirrelationship.Thisalso leadsustosomeinterestingquestionsaboutthetopologyof L m1 ;L m2 and L m1 ballswhich wewillexamineinchapter6.4.2.2NicenessPropertiesofRealizationSubspacesProperty4.4 A k m matrix B issaidtobe\nice"ifitsrowsformanorthogonalbasis withthebasisvectorsallhaving L 2 normequalto p m=k anditscolumnsallhavea L 2 normatmost1.Property4.5 A k -dimensionalrealization B of M issaidtobe almost-spherical ifthe one-normofallunitvectorsin B isatleast p m=k . Property4.6 A k -dimensionalrealization B of M isanicesubspaceiftheinnity-norm ofallunitvectorsinthesubspaceisatmost p k=m . Afterhavingdenedthenicenesspropertiesof B ,weshowtheirequivalence. Property 4.6 isstrongerthanAlmost-sphericalProperty. Weprovethisinthe followingtheorem:Theorem4.7 Forasubspace B of R m , B hasproperty 4.6 implies B isalmost-spherical. Proof Forallvectors w 2 B ,if jj w jj 2 =1and jj w jj 1 p k=m then w needstohave atleast m=k non-zeroelements.Ofthese,the w 'shavingminimumone-normwillhave exactly m=k non-zeros.Forsuch w 'sthecondition jj w jj 2 =1issatisedonlywhenall thenon-zeroshavethesamevaluewhichis p k=m andhence jj w jj 1 = m k : p k=m = p m=k . Thereforeitfollowsthatif jj w jj 2 =1and jj w jj 1 p k=m then jj w jj 1 p m=k .Hence proved.

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17 Property 4.4 property 4.6 . Weshowtheequivalenceofthetwopropertiesbyrst showingthatproperty 4.4 impliesproperty 4.6 ,andthenshowingthatproperty 4.6 implies property 4.4 . Lemma4.8 Forasubspace B of R m , B hasproperty 4.4 ) B hasproperty 4.6 . Proof If B hasproperty 4.4 ,thenthereexistsanorthogonalbasisfor B withthe L 2 norm ofallrowsequalto p m=k andthe L 2 normofallcolumnsatmost1.Therefore,forall unitvectors w 2 B constructedaslinearcombinationsofthebasisvectors( w = a B ,where jj a jj 2 = p k=m ),itholdsthat w ( j )= h a ;B j ijj a jj 2 jj B j jj 2 = p k=m .Thereforeallunitvectorsin B haveaninnitynormatmost p k=m .Thisimpliesthatthesubspace B has property 4.6 . Lemma4.9 Forasubspace B of R m , B hasproperty 4.6 ) B hasproperty 4.4 . Proof Consideranorthonormalbasis(i.e.a k m matrixwithorthonormalrows)for B . Ifallunit-normvectorsin B haveaninnitynormatmost p k=m ,thenthesameholds trueforthebasisvectorsaswell.Let a 2 R k beaunitvector.Now,thevector a B isaunit vectorin B andhencealsohastheaboveproperty.Inotherwords, h a ;B j i p k=m for j 2f 1 ::m g .Alsowehavethat h a ;B j ijj a jj 2 jj B j jj 2 .Sincethepropertyholdstruefor all a 2 R k ,italsoholdstrueforthecasewhen h a ;B j i = jj a jj 2 jj B j jj 2 .Forsuch a ,itholdsthat h a ;B j i = jj a jj 2 jj B j jj 2 p k=m .Since jj a jj 2 =1,itimpliesthat jj B j jj 2 p k=m forall j . Nowifwescalethematrix B bythefactor p m=k then B becomesa k m matrixhaving orthogonalrowswhose L 2 normis p m=k andcolumnswhose L 2 normisatmost1. Theorem4.10 Fora k -dimensionalsubspace B of R m ,property 4.6 isequivalenttoproperty 4.4 . Proof Fromlemma 4.8 andlemma 4.9 ,itfollowsthatfora k -dimensionalsubspace B of R m ,property 4.4 andproperty 4.6 areequivalent. Corollary4.11 Forasubspace B of R m ,property 4.4 almost-sphericalproperty. 4.2.3RephrasingForsterinTermsof B WearenowreadytorephraseForster'sproofofthelowerboun dresultintermsofthe B matrix.

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18 Lemma4.12 Givenanorthogonalbasis B fortherealizationsubspaceof M 2f 1 ; 1 g n m andvectors w i = p k=m u i B;i 2f 1 ::n g suchthatsign ( w i )= M i ,then X i h w i ;M i i 2 = X i k w i k 21 k k M k 2 (4.9) Proof Forsterprovedtheupperboundonhisbridgingquantityas n X i =1 0@ X j jh u i ; v j ij 1A 2 m jj M jj 2 Sincethe B matrixhasthe v j 'sasitscolumns,thereforewehavethat h u i ; v j i = w i ( j ) = p k=m . Substitutingthisintheaboveinequality,weget n X i =1 0@ X j j w i ( j ) j p k=m 1A 2 m jj M jj 2 whichisequivalentto m k n X i =1 jj w i jj 21 m jj M jj 2 Sincesign( w i )=sign( M i ),therefore jj w i jj 1 = h w i ;M i i .Hence n X i =1 h w i ;M i i 2 = n X i =1 jj w i jj 21 k jj M jj 2 ThefollowinglemmacorrespondstoForster'sobservationa bouttheexistenceofa\nice" realization.Lemma4.13 Givena k m matrix B constructedfromthe v j 's,thereexistsanon-singular lineartransformation A 2 GL ( k ) ,suchthatthematrix B 0 = AB cutsthroughthesameset oforthantsandadditionallyhasproperty 4.4 . Proof Forsterprovesthatgiven u i ; v j 2 S k 1 ,thereexistsanon-singularlineartransformation A 2 GL( k )suchthatif u 0 i = A 1 u i and v 0 j = A v j ,then sign h u i ; v j i =sign h u 0 i ; v 0 j i (4.10) and X j v 0 j v 0 j | = m k I k (4.11)

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19 Furthermore,sincescalingdoesnotaectthesignoftheinn erproduct,therefore u 0 i ; v 0 j canbenormalizedsothat u 0 i ; v 0 j 2 S k 1 . Weobserveherethatsincethematrix B hasthe v j 'sasitscolumns,thereforethe matrix B 0 = AB hasthe v 0 j asitscolumns.Alsothematrix P j v 0 j v 0 |j isinfact B 0 B 0 | . Using( 4.11 ),weget B 0 B 0 | = m k I k whichimpliesthat B 0 isanorthogonalmatrixwitheachrowhavinga2-normof p m=k andallcolumnsbeingunit-norm.Toshowthat B 0 realizes M ,weconstruct w 0 i = u 0 i B 0 ;i 2f 1 ::n g .Thenfrom( 4.10 ),itis obviousthatsign( w 0 i ( j ))=sign h u 0 i ; v 0 j i =sign h u i ; v j i =sign( M ij ). Remark Forster'snicerealizationhasproperty 4.4 andisthereforealmost-spherical. Afterhavingestablishedtheexistenceofa\nice"realizat ionsubspaceof M ,werestate Forster'sleftinequalityinthefollowinglemma:Lemma4.14 Givena\nice" B havingproperty 4.4 ,constructunitvectors w i = p k=m u i B asthelinearcombinationoftherowsof B .Then, n m k X i k w i k 21 = X i h w i ;M i i 2 (4.12) Proof Intheprevioussectionweshowedthatforarealizationsubs pace B ,property 4.4 implies B isalmost-spherical.Thismeansthatforallvectors w 2 B; jj w jj 2 =1implies jj w jj 1 p m=k .Nowsincethevectors w i areunitvectors,therefore jj w i jj 1 r m k or, jj w i jj 21 m k or, X i jj w i jj 21 n m k

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20 Theorem4.15 Ifamatrix M 2f 1 ; 1 g n m canberealizedbya k -dimensionalsubspace, then k p mn k M k (4.13) Proof From( 4.9 )and( 4.12 ),weget n m k X i k w i k 21 k k M k 2 (4.14) whichgives, n m k k k M k 2 or, k p mn k M k 4.2.4RephrasingtheGeneralizedLowerBoundinTermsof B Theproofofthegeneralizedlowerboundintermsofthe B matrixisverysimilarto theproofofForster'soriginalresult.Themaindierencei sintheinterpretationofthe bridgingquantityusedinthetworesults.Whilebothuse n X i =1 h w i ;M i i 2 asthebridgingquantity,Forster'soriginalresultholdso nlyfor M 2f 1 ; +1 g n m and thereforethebridgingquantityequals P i jj w i jj 21 whichisnotthecaseforthegeneralized lowerbound.Soforthegeneralizedproof,wekeepthebridgi ngquantityinitsoriginalform andprovetheupperboundonitas X i h w i ;M i i 2 k jj M jj 2 Thenweprovethelowerboundonthebridgingquantitywhichh oldsforalmostsphericalrealizationsubspaces.Thelowerboundis X i h w i ;M i i 2 n m k (min i;j j M ij j ) 2

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21 Theproofforboththeseresultsisadirectadaptationofthe generalizedlowerbound proofin[ 4 ]obtainedbysubstituting h u i ; v j i = w i = p k=m .Combiningthetwoinequalities, wegetthegeneralizedlowerbound.4.2.5IntuitiveJustication Therightinequalityin( 4.14 )istrueforallrealizationsubspacesof M ,whereasthe leftinequalityistrueforonlythe\nice"subspacesthatha veproperty( 4.4 ).Thusboth inequalitiessimultaneouslyholdonlyforthe\nice" B 's. ThereasonforrephrasingForster'sinequalitiesinthisma nneristhatthebridgingquantity intheForster'sresultvariedwiththedimension k oftherealizationwhichappearstobe counter-intuitive.Inourformulation,the w i 'sareallunit-normvectorsandhenceitensures thatthebridgingquantityremainsthesameirrespectiveof therealizationdimension.We alsointroduce k intotherightinequalitysothattherightsidequantitynow increaseswith k insteadofbeingindependentof k asitwasinForster'sformulation.

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CHAPTER5 NICEREALIZATIONS Theexistenceofnicerealizations-realizationsthathave somenon-trivial`nice'properties-helpsinobtainingbetterboundsiftheirexistence canbeproven.Thistechnique wasusedbyForstertoobtainhislowerboundresult.Inthiss ection,wedeneanother `nice'propertyofourrealizationandrelateitwiththeone sdenedinthepreviouschapter. Wemakeaconjectureabouttheexistenceofsubspacesthatha veboththe`nice'properties whichifprovenresultsinimprovedbounds. 5.1NewDenitionofNiceness Property5.1 A k -dimensionalrealization B of M isniceifforall i ,theprojectionofthe vectordenedbyrow M i on B liesinthesameorthantas M i . Tomotivatetheintroductionofthisnewnotionof`niceness ',weshowimprovedbounds forForster'sresultandthegeneralizedlowerboundresult assumingtheexistenceofnice realizationshavingproperty 5.1 . Conjecture Thereexistsarealization B of M ,thatisalmost-sphericalandalsohasproperty 5.1 . 5.1.1ImprovedBoundsfor M 2f 1 ; +1 g n m Lemma5.2 Givena\nice"realization B of M 2f 1 ; +1 g n m thathasproperty 5.1 ,let vectors w i beunitvectorsinthedirectionof M i projectedon B .Then, jj w i jj 1 = vuut k X l =1 h M i ;B l i 2 where B l isthe l throwof B . Proof For B torealize M ,thereshouldexistvectors w i ;i 2f 1 ::n g suchthatsign( w i )= M i . Sincethesubspace B hasproperty( 5.1 ),thereforetheprojectionof M i on B liesinthe sameorthantas M i .Hencethechoiceof w i asbeingaunitvectorinthedirectionof M i 22

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23 projectedonto B isperfectlyvalid. The j thco-ordinateofvector w i is w i ( j )= P kl =1 h M i ;B l i :B lj p P l h M i ;B l i 2 Now jj w i jj 1 = X j j w i ( j ) j = h w i ;M i i = m X j =1 w i ( j ) M ij or, jj w i jj 1 = m X j =1 P l h M i ;B l i :B lj p P l h M i ;B l i 2 M ij or, jj w i jj 1 = P l h M i ;B l i 2 p P l h M i ;B l i 2 = s X l h M i ;B l i 2 Henceproved. Lemma5.3 Forarealization B of M 2f 1 ; +1 g n m thathasproperty 5.1 , X i jj w i jj 1 2 = X i X 1 l k h M i ;B l i 2 nk + k m : ORT ( M ) where ORT ( M )=2 X j 1 ;j 2 h M j 1 ;M j 2 i forall j 1
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24 and X l B lj :B lj 0 = X l B | jl :B lj 0 = B | B ( j;j 0 ) : Thereforetheexpressionreducesto X j X j 0 M | M ( j;j 0 ) B | B ( j;j 0 ) or X j ( jj M j jjjj B j jj ) 2 +2 : X h M j 1 ;M j 2 ih B j 1 ;B j 2 i (forall j 1
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25 Theorem5.5 Ifamatrix M 2f 1 ; +1 g n m canberealizedbya k -dimensionalsubspace then k s mn n + ORT ( M ) m Proof Thisresultrequirestheexistenceofarealizationsubspac ethatisalmost-spherical andalsohasproperty 5.1 ,whichweconjecturedtobethecase. Since B isalmost-spherical,thereexistunitvectors w i suchthatsign( w i )= M i and n X i =1 jj w i jj 21 n m k ; fromlemma 4.14 . Also,inlemma 5.3 weshowedthatif B hasproperty 5.1 ,then n X i =1 jj w i jj 21 nk + k m : ORT( M ) Therefore,forarealizationsubspacethatisalmost-spher icalandalsohasproperty 5.1 ,it holdsthat n m k n X i =1 jj w i jj 21 nk + k m : ORT( M ) Thisgives, n m k k n + ORT ( M ) m Thisdirectlygivesthestatedboundfor k . Corollary5.6 ForaHadamardmatrix H n ofsize 2 n 2 n ,theminimaldimensionofa realizationsubspaceisboundedas k r 2 n : 2 n 2 n +0 =2 n= 2 5.1.2ImprovedBoundsfor M 2 R n m Lemma5.7 Givena\nice"realization B of M 2 R n m thathasproperty 5.1 ,letvectors w i beunitvectorsinthedirectionof M i projectedon B .Then, m X j =1 M ij w i ( j )= h w i ;M i i = vuut k X l =1 h M i ;B l i 2 where B l isthe l throwof B .

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26 Proof Inthiscasetoothechoiceof w i astheunitvectorsinthedirectionof M i projected onto B isperfectlyvalidsince B hasproperty 5.1 . Again,the j thco-ordinateofvector w i is w i ( j )= P kl =1 h M i ;B l i :B lj p P l h M i ;B l i 2 Now h w i ;M i i = m X j =1 w i ( j ) M ij = m X j =1 P l h M i ;B l i :B lj p P l h M i ;B l i 2 M ij or, h w i ;M i i = X l h M i ;B l i P j M ij B jl p P l h M i ;B l i 2 or, h w i ;M i i = X l h M i ;B l i 2 p P l h M i ;B l i 2 = s X l h M i ;B l i 2 Henceproved. Lemma5.8 Forarealization B of M 2 R n m thathasproperty 5.1 , n X i =1 h w i ;M i i 2 = n X i =1 X l h M i ;B l i 2 k m 0@ X j ( jj M j jj ) 2 +2 : ORT ( M ) 1A Proof X i X 1 l k h M i ;B j i 2 = X 1 l k X i ( X j M ij :B lj ) 2 = X i X 1 l k X j X j 0 M ij B lj M ij 0 B lj 0 = X i X j X j 0 M ij :M ij 0 X l B lj :B lj 0 = X j X j 0 X i M ij :M ij 0 X l B lj :B lj 0 Now, X i M ij :M ij 0 = X i M | ji :M ij 0 = M | M ( j;j 0 ) and X l B lj :B lj 0 = X l B | jl :B lj 0 = B | B ( j;j 0 ) :

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27 Thereforetheexpressionreducesto X j X j 0 M | M ( j;j 0 ) B | B ( j;j 0 ) or X j ( jj M j jj 2 jj B j jj 2 ) 2 +2 : X h M j 1 ;M j 2 ih B j 1 ;B j 2 i (forall j 1
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28 Thisgives, M 2 min n m k k m 0@ X j jj M j jj 22 + ORT ( M ) 1A or, k 2 M 2 min m 2 n P j jj M j jj 22 + ORT ( M ) Thestatedboundon k directlyfollows. Thuswehaveshownthattheexistenceofnicerealizationsgi vesusanewboundon theminimaldimension.5.1.3WhyIsThisBoundBetter? Thenormofthematrix M thatappearsinthedenominatorofForster'sboundis denedas jj M jj =sup x 2 R m jj M x jj 2 jj x jj 2 Nowif x =(1 ; 1 ; 1 ;:::; 1)beoneofthevectorsthatcausesthesupremumtobeattaine d, thenwehave jj M jj = jj M x jj 2 jj x jj 2 = q P i ( P j M ij ) 2 p m = q P j jj M j jj 22 + ORT ( M ) p m whichisthedenominatorofournewbound(notethat P j jj M j jj 22 = mn forthecasewhen M 2f 1 ; 1 g n m ).ItthereforefollowsthatourboundisasgoodasForster's whenthe all-onesvectorisoneofthesupremizingvectors(suchasfo rHadamardmatrices)andin allothercasesourboundisbetterbecauseourdenominatori slesserthan jj M jj . 5.2NicePropertiesofSpecicConstructions InthissectionweshowthattheconstructionsgivenbyForst erinhisdoctoraldissertation[ 6 ] andbyBelcher etal in[ 7 ]haveproperty 5.1 . 5.2.1Forster'sConstruction Forster[ 6 ]gavetheconstructionofarank3matrix M 0 thathadthesamesignpattern as H 2 .Heusedtheconstructiontogiveanupperboundof3 d n 2 e ontherealizationdimension.

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29 Forthe2 n 2 n Hadamardmatrix,thematrix M 0 2 n constructedas: M 0 2 n = M 0 N n 2 = 0BBBBBBB@ 11111 15 1 15 1 1 1 1 15 1CCCCCCCA N n 2 (5.1) isofrank3 n 2 . 5.2.2Forster'sconstructionhasproperty 5.1 ToshowthatForster'sconstructionhasproperty 5.1 ,wewilluseanorthonormalbasis B for M 0 suchthattheprojectionof M i on B liesinthesameorthantas M i . Lemma5.11 Therowsofthematrix B givenby 0BBBB@ 1 = 21 = 21 = 21 = 2 01 = p 2 1 = p 20 01 = p 61 = p 6 2 = p 6 1CCCCA formanorthonormalbasisfor M 0 . Usingthisbasis,weshowbyinductionthatForster'sconstr uctionhasproperty 5.1 . Lemma5.12 BaseCase:For M = H 2 2 , w i = M i j B liesinsameorthantas M i . Proof Given, B = 0BBBB@ 1 = 21 = 21 = 21 = 2 01 = p 2 1 = p 20 01 = p 61 = p 6 2 = p 6 1CCCCA ; M = 0BBBBBBB@ 11111 11 1 11 1 1 1 1 11 1CCCCCCCA Now, w i = M i j B = P j h M i ;B j i :B j .Therefore, w 1 =(1111) w 2 =(0 2 = 34 = 3 2 = 3) w 3 =(04 = 3 2 = 3 2 = 3) w 4 =(0 2 = 3 2 = 34 = 3) Itisobviousthat w i liesinthesameorthantas M i .

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30 Lemma5.13 InductionStep:Iffor M k = H 2 k ; w i k = P j h M i k ;B j k i :B j k liesinthesame orthantas M i k ,thenfor M k +1 = M k N M; w i k +1 = P j h M i k +1 ;B j k +1 i :B j k +1 liesinthe sameorthantas M i k +1 . Proof The i throwofthematrix M k +1 , M i k +1 iseither M i k N M 1 or M i k N M 2 orsoon. Considerthecasewhen M i k +1 = M i k N M 1 .Then, w i k +1 = X j h M i k +1 ;B j k +1 i :B j k +1 or, w i k +1 = X j X l h M i k O M 1 ;B j k O B l i :B j k O B l or, w i k +1 = X j X l h M i k ;B j k ih M 1 ;B l i :B j k O B l or, w i k +1 = X j h M i k ;B j k i :B j k OX l h M 1 ;B l i :B l or, w i k +1 = w i k O w 1 Sincetensorproductpreservessignpatterns,thereforewe cansaythatif w i k hasthesame signas M i k andif w 1 hasthesamesignsas M 1 ,then w i k +1 = w i k N w 1 willhavethe samesignsas M i k +1 = M i k N M 1 . Similarargumentcanmadeaboutthecaseswhen M i k +1 iseither M i k N M 2 or M i k N M 3 or M i k N M 4 . Theorem5.14 Forall n ,Forster'srealizationof M n = H 2 n hasproperty 5.1 . Proof Theresultfollowsfromthebasecase(lemma 5.12 )andtheinductionstep(lemma 5.13 ).

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31 5.2.3Belcher-Hicks-SitharamConstruction Thisconstructiongivesanupperboundof3 = 4 2 p ontherealizationdimensionof H 2 p .The M 0 thatrealizes H 2 2 isgivenby: 0BBBBBBB@ 11131 31 1 11 3 1 3 1 11 1CCCCCCCA andisofrank3.The M 2 k +1 isconstructedfrom M 2 k asfollows:Therst2 k rowsof M 2 k +1 areconstructedas( M 2 k ( i ) M 2 k ( i )) ; 1 i 2 k .Thenext2 k rowsareconstructedas ( M 2 k ( i ) M 2 k ( i )) ; 2 k +1 i 2 k +1 . 5.2.4Belcher-Hicks-Sitharamconstructionhasproperty 5.1 Againtoshowthatthisconstructionhasproperty 5.1 ,wewilluseanorthonormal basis B for M 0 suchthattheprojectionof M i on B liesinthesameorthantas M i . Lemma5.15 Therowsofthematrix B givenby 0BBBB@ p 3 = 6 p 3 = 6 p 3 = 6 p 3 = 2 01 = p 2 1 = p 20 2 = p 61 = p 61 = p 60 1CCCCA formanorthonormalbasisfor M 0 . Usingthisbasis,weshowbyinductionthatthisconstructio nhasproperty 5.1 . Lemma5.16 BaseCase:For M = H 2 2 , w i = M i j B liesinsameorthantas M i . Proof Given, B = 0BBBB@ p 3 = 6 p 3 = 6 p 3 = 6 p 3 = 2 01 = p 2 1 = p 20 2 = p 61 = p 61 = p 60 1CCCCA ; M = 0BBBBBBB@ 11111 11 1 11 1 1 1 1 11 1CCCCCCCA

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32 Now, w i = M i j B = P j h M i ;B j i :B j .Therefore, w 1 =(3 = 23 = 23 = 29 = 2) w 2 =(1 = 2 3 = 21 = 2 1 = 2) w 3 =(1 = 21 = 2 3 = 2 1 = 2) w 4 =(3 = 2 1 = 2 1 = 21 = 2) Itisobviousthat w i liesinthesameorthantas M i . Lemma5.17 Inductionstep:Iffor M k = H 2 k ; w i k = P j h M i k ;B j k i :B j k liesinthesame orthantas M i k ,thenfor M k +1 givenby 0B@ M k M k M k M k 1CA w i k +1 = P j h M i k +1 ;B j k +1 i :B j k +1 liesinthesameorthantas M i k +1 ,where B k +1 givenby 0B@ B k B k B k B k 1CA Proof For1 i 2 k , w i k +1 = X j h M i k +1 ;B j k +1 i :B j k +1 or w i k +1 =2 : d= 2 X j =1 h M i k ;B j k i :B j k +1 +0 : d X j = d 2 +1 B j k +1 ( d =3 = 4 2 k +1 ) or w i k +1 =2 : d= 2 X j =1 h M i k ;B j k i : ( B j k B j k )=( w i k w i k ) Since w i k hasthesamesignsas M i k ,thereforethevector( w i k w i k )hasthesamesignsas ( M i k M i k ). For2 k +1 i 2 k +1 , w i k +1 = X j h M i k +1 ;B j k +1 i :B j k +1 or w i k +1 =0 : d= 2 X j =1 B j k +1 +2 : d X j = d 2 +1 h M i k ;B j k i :B j k +1 ( d =3 = 4 2 k +1 )

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33 or w i k +1 =2 : d= 2 X j =1 h M i k ;B j k i : ( B j k B j k )=( w i k w i k ) Since w i k hasthesamesignsas M i k ,thereforethevector( w i k w i k )hasthesamesigns as( M i k M i k ). Thereforetheresultfollows. Theorem5.18 Forall n ,Belcher-Hicks-Sitharamconstructionof M n = H 2 n hasproperty 5.1 . Proof Theresultfollowsfromthebasecase(lemma 5.16 )andtheinductionstep(lemma 5.17 ).

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CHAPTER6 THETOPOLOGYOFBALLS Inchapter4wedenedproperty( 4.5 )-whichwecallthe almost-spherical property -andproperty( 4.6 )forrealizationsubspacesof M .Thesepropertiesnaturallyleadto interestingquestionsaboutthetopologyof L 1 ;L 2 and L 1 balls.Toposeandanswerthose questions,werstformallyrestatethealmost-sphericalp ropertyintermsof L 1 and L 2 balls. 6.1TheGeneralizedAlmost-SphericalProperty Herewegiveageneraldenitionofthealmost-sphericalpro pertyforsubspacesof R m asageneralizationofproperty( 4.5 ). Property6.1 Asubspace B of R m issaidtobe r -sphericalfor r 2 R if ( L m1 = r ) \ B ( L m2 =1) \ B InplainEnglish,thepropertysaysthatasubspace B is r -sphericalifthe L 1 = r ball interesectedwith B isfullycontainedintheintersectionofthe L 2 =1ballwith B . Thisstatementisageneralizedversionofproperty( 4.5 )sinceitsaysthatforall vectorsin w 2 B , jj w jj 1 = r implies jj w jj 2 1orinotherwords, 8 w 2 B; jj w jj 2 =1 implies jj w jj 1 r (whichisproperty( 4.5 )for r = p m=k ). Thequestionweaskandseektoanswerinthischapteris: Whatisthemax. k suchthatthereexistsa k -dim.subspace B of R m thatis r -spherical foragiven r ? Alternatively, given k ,whatisthemax. r suchthatthereexistsa k -dim.subspaceBof R m thatis r -spherical? Beforetryingtoarriveataclosed-formexpressionfor r intermsof m and k ,weprove boundsonthevalueof r forspeciccases.Thegoalistogivearangeinwhichthelimi ting valueof r ,say r lim ,mayliesuchthatfor r r lim thereexistsasubspacethatis r -spherical 34

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35 andfor r>r lim thereexistsnosubspacethatis r -spherical.Thusforeveryspeciccase thatweworkout,weprovethefollowingtwostatements: 1. LowerBoundStatement (implies r lim r lb ): For r r lb ,thereexistsasubspace B ofdim. k ,suchthat 8 w 2 B; ( k w k 1 = r ) ) ( k w k 2 1) Weprovethisstatementforsomeextremevectors w 2 B ,whichareshowntobe sucienttoinspect. 2. UpperBoundStatement (implies r lim r ub ): For r>r ub ,forall k -dim.subspaces B of R m , 9 w 2 B; suchthat( k w k 1 = r )and k w k 2 > 1 i.e.( L m1 = r ) T B isnotcompletelycontainedin( L m2 =1) T B . Afterweprovethetwostatements,wewillhavenarroweddown r lim forthatparticular caseto r lb r lim r ub . Wenoteherethatweonlyneedtoshow r lim p m=k forrephrasingForster'sresultin termsofalmost-sphericalsubspaces. 6.2ObservationsAboutTheGeneralizedAlmost-sphericalP roperty Inthissection,weanswerthequestionposedatthebeginnin gofthechapterfor specicvaluesof m and k togaininsightintothetopologyof L 1 and L 2 ballsandintothe dependenceof r on m and k . Proposition1. For r 1, all subspacesof R m ofanydimension k ( m )are r -spherical. Proposition2. For r> p m ,theredoesnotexist any subspaceof any dimension k that is r -spherical. Proposition3. For r = p m ,only1-dimensionalsubspacesare r -spherical.

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36 Proposition4. For k = m 1,theextremalvalueof r;r lim = p 2. Weshowedthatfor r p 2,wecanconstructa B ? =(1 ; 1 ; 1 ;:::; 1)suchthatthe( m 1)dim.subspaceBis r -spherical,andnoother B ? doesbetter.Inthiscase,thenumberof extremalpointson B isgivenby2 : m 2 . Proposition5. For k = m 2, r lim p 2. Weagainconstructeda B ? spannedbythevectors(1 ; 1 ;:::; 1)and(1 ; 1 ;:::; 1)and showedthatthesubspaceBis r -sphericalfor r p 2.Thoughinthiscase,wecouldn't showthatnoother B ? doesbetter.Therefore,weonlygetalowerboundon r lim .Inthis case,thenumberofextremalpointson B isgivenby2 : m 1 2 . Proposition6(Conjecture). Form=4andk=2, r lim =1+1 = p 2. Proof Whilewecanprovethelowerbound,i.e. r lim 1+1 = p 2,wedonothaveacomplete prooffortheupperbound;onlyaproofoutlinethatwebeliev ewillgiveus r lim 1+1 = p 2. Lowerbound: r lim 1+1 = p 2. Considerthefollowing2-dimensionalsubspaceof R 4 : B = 0B@ 101 = p 2 1 = p 2 011 = p 21 = p 2 1CA Werstnotethattheextremalvectorsin B (i.e.vectorswith L 1 norm r and L 2 normascloseto1aspossible)willhaveasmanyzeroesasposs ible,whichis k 1. Onesuchvectoris(1 ; 1) B =( 1 2 ; 1 2 ; 1 p 2 ; 0)for r =1+1 = p 2.Forsuchextremalvectors x 2 B ,itcanbeshownthatfor r 1+1 = p 2 ; ( jj x jj 1 = r ) ) ( j x jj 2 1). Upperbound(Conjecture): r lim 1+1 = p 2. Whilewedonothaveacompleteprooffortheupperbound,west ronglybelievethe conjecture.Wepresenthereaproofoutlinethatwebelievec anbeextendedtoprove theconjecturedupperbound.Letageneral B ? be, B ? = 0B@ c 1 c 2 c 3 c 4 d 1 d 2 d 3 d 4 1CA

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37 This2 4subspacecanberepresentedasapointona Grassmannmanifold in 42 -dimensionalspace.Theco-ordinatesofthepointaregiven bythe 42 =6determinants: D 12 ;D 13 ;D 14 ;D 23 ;D 24 ;D 34 ,where D ij is, D ij = c i c j d i d j 6 =0 whichsatisfytheGrassman-Pluckerrelation, D 12 D 34 + D 13 D 24 + D 14 D 23 =0(6.1) Now,any x 2 B willbeorthogonalto B ? .Hence, 0B@ c 1 c 2 c 3 c 4 d 1 d 2 d 3 d 4 1CA 0BBBBBBB@ x 1 x 2 x 3 x 4 1CCCCCCCA =0(6.2) Additionally, x hasa1-normequalto r , X i j x i j = r (6.3) andshouldhavea2-normlessthanorequalto1, s X i x i 2 1(6.4) Sinceweknowthattheextremalvectorsin B willhave k 1zeros,wesetoneofthe x i 'sinturntozeroandget4inequalitiesfromthe2-normcondi tion ( D 12 ) 2 +( D 13 ) 2 +( D 23 ) 2 1 r 2 ( j D 12 j + j D 13 j + j D 23 j ) 2 (6.5) ( D 12 ) 2 +( D 14 ) 2 +( D 24 ) 2 1 r 2 ( j D 12 j + j D 14 j + j D 24 j ) 2 (6.6) ( D 13 ) 2 +( D 14 ) 2 +( D 34 ) 2 1 r 2 ( j D 13 j + j D 14 j + j D 34 j ) 2 (6.7)

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38 ( D 23 ) 2 +( D 24 ) 2 +( D 34 ) 2 1 r 2 ( j D 23 j + j D 24 j + j D 34 j ) 2 (6.8) Thus,forthesix D ij 'storepresentavalid2-dim. r -sphericalsubspaceof R 4 ,they havetosatisfythe4inequalitiesandtheGrassmann-Plucke rrelationship. Now,withoutlossofgenerality,weassumethat j D 34 jj D 12 j ,andso j D 34 j isatleast fourthhighestintheorderingofthe6determinants.Weclaimthatthemaximumvalueof r thatsatisesall4inequalitiesatthesametime isattainedwhenthe D ij 'stakevaluesthatareasclosetoeachotheraspermittedby theGPrelation.Thismaximumvalueisattainedbyaparticul ar\extremal"ordering ofthe j D ij j 's. Assumption: Letusassumethefollowingorderingof D ij 's: j D 13 jj D 14 jj D 23 jj D 24 jj D 12 jj D 34 j (6.9) Theobjectiveistoshowthatthemaximumvalueof r overallvalid D ij 'sforwhich allfourinequalitiessimultaneouslyholdis1+1 = p 2. Also,( 6.1 )canbewrittenas j D 12 D 34 j = j D 13 D 24 + D 14 D 23 j (6.10) From( 6.9 )and( 6.10 )wenotethat, j D 34 j p 2 j D 13 j From( 6.7 ),weobservethatoverallpermissiblevaluesof D ij 's,theexpression s ( j D 13 j + j D 14 j + j D 34 j ) 2 ( D 13 ) 2 +( D 14 ) 2 +( D 34 ) 2 ismaximizedwhen j D 34 j = p 2 j D 13 j and j D 14 j = j D 13 j andisequalto1+1 = p 2. Nowforthisordering,ifthemaximumvaluesfor r obtainedfrom( 6.5 ),( 6.6 )and ( 6.8 )aregreaterthan1+1 = p 2then r 1+1 = p 2isthevaluethatsatises allfour inequalities.Andweknowthatforthefollowingassignment , allfour inequalities

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39 attain r 1+1 = p 2: j D 13 j = j D 14 j = j D 23 j = j D 24 j = j D 12 j = p 2= j D 34 j = p 2 Thus,forthisordering( 6.7 )actsasthe\bottleneck"inpreventingtheupperbound on r togoabove1+1 = p 2. Weclaimthatforeachordering,onecanndaninequalityana logousto( 6.7 )that willactasabottleneckforthatordering.Wefurtherclaimt hatnoneoftheother orderingscandobetterthanthis,inotherwordsallthetota lorderingscorresponding tothepartialorderthatgivesus j D 34 j p 2 j D 13 j areextremal. Basedonourexperienceworkingoutthesecases,wemakethef ollowingremarks: 1.Theexercisehereistopicktheinequality/inequalities thatgive(s)thetighestupper boundonthevalueof r . 2.Attheextremalvalueof r ,thenumberofinequalitiesthatturnintoequalitiesisequ al tothenumberofvariables(i.e.,the6 D ij 'sinthe m =4, k =2case). 3.Weconjecturethatregardlessofthevalueof m and k ,weonlyneedtolookataxed numberoforderingstoobtainthetightestupperboundon r . 4.ForinsightsintotheForsterproblem,onlya lower boundon r lim isneeded.But wewouldhavetounderstandsomethingaboutthefamilyofcon structions/subspaces thatestablishthislowerbound{i.e,beabletosaythatatle astoneofthemcanbe madetocutthroughanysetoforthantsthatanarbitrarysubs paceof k dimensions can.Notethatwithournewapproach,aconstructionisjusta settingofvaluesfor the D ij . 6.3ObservationsontheFigure Aftercomputingthelimitingvalueof r forspecicvaluesof m and k andobserving itsvariationwith m and k ,wemakethefollowingobservations: 1. r ( m +1 ;k ) >r ( m;k ).Theextremalvalueof r increaseswith m foraconstant k . 2. r ( m;k ) r ( m;k +1).Theextremalvalueof r decreaseswithincreasein k for aconstant m .Thishappensbecausethenumberofindependentcolumnsin B ? decreasesandhenceitbecomeseasiertonda\bad" x .

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40 r(m,1) = sqrt(m) k=m-2 k=m-1 k = m < < < < < < 65 4321 6m>= >= >=<<>=r = sqrt(8/3)< <
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41 3. r ( m;m d ) r ( m +1 ;m d +1).Foraconstantdierencebetween m and k ,the extremalvalueof r ismonotonicallynonincreasing. Webelievethattheclosed-formexpressionfor r willlooklike r ( m;k )= p m a p k forsomeconstant a ,thoughwefoundithardtoprovethatthisisthecase.

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CHAPTER7 CONCLUSIONANDFUTUREWORK Wehavepresentedthestudyofageometriclowerboundquesti on:whatistheminimal dimensionofasubspacethatintersectsagivensetoforthan ts.Westartwithits`avatar' asalongstandingopenproblemincommunicationcomplexity anditsrecentsolutionwhere theproblemwasstatedasthatofndingalineararrangement ofhomogeneoushalfspaces inthelowestdimension.Weexplainwhythetreatmentofthep roblemanditsresultseem counterintuitiveandjustifyrephrasingitasndingthemi nimaldimensionofasubspace thatintersectsagivensetoforthants.Thisrephrasinglea dsustosomeinteresting`nice' propertiesoftheserealizationsubspacesthatenableusto showbetterbounds.Wealso explainwhythenewboundsbearoutourinitialbewilderment overtheoriginalresult, namelythattheminimaldimensionofarealizationsubspace shouldbeafunctionofthe setoforthantsitsrealizesandnotjustsomealgebraicquan titysuchas jj M jj .Thealmostsphericalpropertyoftheserealizationsubspacesleadsus tosomequestionsinfunctional analysisandmetricembeddingssuchas:whatisthelimiting value r suchthatthe L 1 = r balliscompletelycontainedinthe L 2 =1ballin R m . Wewouldalsoliketoidentifysomedirectionstoproceedfur theronthisproblemand alsosomeconnectionswithproblemsinotherareasthatcoul dbeexploredinthefuture. 1.Thereareanumberofquestionsabout`niceness'ofrealiz ationsubspaces-canwe deneotherkindsofniceness?Canweshowthatifthereexist sarealization,then thereexistsarealizationthatisniceinthenewlydenedse nse?Aredierentkinds of`niceness'-esequivalent?Dotheyhelpusattaintighter bounds?Forexample, (a)Doesmaintainingmaximumdistancebetweenthecolumnso f B haveanysignificanceasanicepropertythathelpstoattainbetterresults ?Or (b)Doeshavingcolumnsof B splitinto k bundlessuchthateachbundleisnearly orthogonaltoeveryotherqualifyasa`nice'propertyofany use? 42

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43 2.Supposetheinnerproductsofeverypairof(asubsetof l )rowsof M isbounded belowbysome .Whatdoesitsayaboutthenormof M (intermsof l and )?Can weusethisseparationasthepropertyof M weareinterestedininsteadofthenorm? Ordoesitturnouttobeequivalenttothenorminsomesense?D oesitgiveabetter boundthantheoneweobtainedusing ORT ( M )? 3.Thoughweunderstoodthevariationof r with m and k forthegeneralizedalmostsphericalpropertyinchapter6,wewereunabletocomeupwit haclosed-formexpression.Isitpossibletoobtainaclosed-formexpression for r intermsof m and k ? Thisleadstoanumberofquestionstryingtoconnectourresu ltwithsomeresults fromfunctionalanalysisandtheoryofnormedspaces.Forex ample, (a)IthasbeenshownbyV.Milman etal [ 8 ]thatintherightprobabilityspace, most k -dimensional( k>ca 2 m ,forsomeuniversalconstant c )subspaces B of R m havethepropertythatthe L 1 normofallunitvectors w in B isboundedas ( r 2 + a ) p m jj w jj 1 ( r 2 + a ) p m Canweshowtheexistenceofone k -dimensionalsubspacethathastheabove propertyandalsohasoneofthe`nice'propertiesweuse?Doe sthatimprovethe bound? (b)ThereisacelebratedtheorembyDvoretzkyaboutembeddi ngof L p normed spacesinto L 2 ,ofwhichoursisthespecialcasefor p =1.Beforewestatethe theorem,wedenethe Banach-Mazurdistance betweentwonormedspaces X and Y tobe c ,ifthereisa linear map f : X ! Y withdistortion( f ) c . Dvoretzky'stheoremstatesthatTheorem7.1(ByDvoretzky) Forevery n and > 0 ,every n -dimensional normedspacecontainsa k =n( log n ) -dimensionalspacewhoseBanach-Mazur distancefrom L 2 is 1+ . Theproofofthistheoremisprobabilisticinnature.Isitpo ssibletogivea constructiveproofofthiscelebratedtheorem? (c)Isitpossibletocharacterizethefamilyofall k -dimensionalsubspacesthatare r -spherical?

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44 4.Thereareanumberofquestionsaboutcell(orthant)-equi valenceofsubspacesand lineartransformationsthatpreservecell-equivalence.F orexample, (a)Whenare2 k -dim.cellcomplexes(acellcomplexisnothingbutasetofor thants) with m hyperplanes combinatorially cellequivalent(i.e.havethesamesetof cells)?Inotherwords,whenare2 k -dim.subspacesof m -spacecombinatorially cellequivalent? (b)Whenare2 k -dim.cellcomplexeswith m hyperplanes real cellequivalent(i.e. thenormalstothehyperplanesforoneshouldbeobtainablef romthatofthe otherbyatransformationthatpreservesangles)? (c)Toanswerboththeabovequestions,characterizeacompl etesetof( GL ( m ){ lineartransformationsin m -space)operationsthatcanbeperformedtoa k m matrixthatwillpreservecombinatorial(respectivelyrea l)cellequivalence.Are alloftheseoperationscontainedin GL ( r )forsome r m ? (d)Givenacellcomplex C with m hyperplanesin k -space,howtogeta k -dim. subspaceof m spacethatiscombinatoriallycell-equivalentorrealcell -equivalent to C ?

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REFERENCES [1]JurgenForster.\ALinearLowerBoundontheUnboundedE rrorProbabilisticCommunicationComplexity" JournalofComputerandSystemSciences ,vol.65,pp. 612{625,2002. [2]R.PaturiandJ.Simon.\ProbabilisticCommunicationCo mplexity," Journalof ComputerandSystemSciences ,vol.33,pp.106{123,1986. [3]N.Alon,P.FranklandV.Rodl.\GeometricalRealizatio nofSetSystemsandProbabilisticCommunicationComplexity,"in Proceedingsofthe26thAnnualSymp.on FoundationsofComputerScience(FOCS) ,pp.277{280,PortlandOR,1985. [4]JurgenForster,MatthiasKrause,SatyanarayanaV.Lok am,RustamMubarakzjanov, NielsSchmittandHans-UlrichSimon.\RelationsBetweenCo mmunicationComplexity,LinearArrangements,andComputationalComplexi ty,"in Proceedingsofthe 21stConferenceonFoundationsofSoftwareTechnologyandT heoreticalComputer Science ,pp.171{182,2001. [5]EranHalperin,JeremyBuhler,RichardKarp,RobertKrau thgamerandBenWestover.\DetectingProteinSequencesViaMetricEmbeddings, "in Proceedingsofthe 11thInternationalConferenceonIntelligentSystemsforM olecularBiology(ISMB 2003) 122-199,Brisbane,Australia,2003. [6]JurgenForster.\SomeResultsConcerningArrangement sofHalfSpacesandRelative LossBounds," Ph.DDissertation,LehrstuhlfurMathematikundInformat ik,Bochum, Germany ,2001. [7]M.Belcher,S.Hicks,M.Sitharam.\Equiseparations,"i n ACM-SE39:Proceedings ofthe39thAnnualSoutheastRegionalConference ,ACMPress,AthensGA(2001). [8]V.D.Milman,G.Schechtman.\AsymptoticTheoryofFinit e-dimensionalNormed Spaces,"in LectureNotesinMathematics1200, Springer-Verlag,Berlin(1986). [9]PiotrIndyk.\AlgorithmicAspectsofGeometricEmbeddi ngs,"in Proceedingsofthe 42ndAnnualSymposiumonFoundationsofComputerScience ,pp.10{35,LasVegas, 2001. [10]N.Linial.\FiniteMetricSpaces-Combinatorics,Geom etryandAlgorithms,"in ProceedingsoftheInternationalCongressofMathematicia nsIII ,pp.573{586,Beijing, 2002. 45

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BIOGRAPHICALSKETCH PranavDandekarwasborninIndore,India,in1980.Hewentto schoolinChoithram HighSchool,Indore(1983-1995),andinSriSathyaSaiVidya Vihar,Indore(1995-1997). Hecompletedhisundergraduatestudiesbetween1998and200 2fromShriG.S.Institute ofTechnologyandScience,Indore,whichisaliatedtoRaji vGandhiTechnicalUniversity(MadhyaPradesh),India.HegraduatedwithaB.Enginco mputerengineeringwith distinctioninAugust2002.Hespentthefallof2002working asaresearchassistantat KReSIT,IITBombay,ondatabaseperformancebenchmarkinga ndoptimization.Hehas beenagraduatestudentinCISEattheUniversityofFloridas inceJanuary2003.HeinternedwithAmazom.cominSeattle,WA,duringthesummerof2 004andintendstogo backtheretoworkfull-timeaftergraduatingfromtheUnive rsityofFlorida. 46