PDF-isthe\sumofsquaresofx"andSyy=nXi=1(y�y)2=nXi=1y2i�1

n nXi1yi2isthesumofsquaresofyThevalueofrxyisalwayssuchthat0120rxy201Weonlyobtainrxy61ifthepointsonascatterplotlieexactlyalongastraightlineNote1Noneofthisimpliescausality2Wecanca. EveryvectorinS0canbewrittenasalinearcombinationofthebasisvectorsofS0:=nXi=1ii:(2)UsingEq.(1),weobtaing=nXj=1nXi=1Dji(g)ij:(3)Fromthisresult,weseethatwecanconsidergtooperateonthecoecientsirat case:E[s2]=E[1 NNXi=1(xi x)2]=1 x+NXi=1 x2]WeknowPNi=1xi=N
xandPNi=1 x2=N
x2.Plugtheseintothederivation:E[s2]=1 NE[NXi=1x2i2N
x2+N
x2]=1 NE[NXi=1x2iN
x2]=1 NE[NXi=1x2i]E[ x2]=E[x2]E[ x2]Acco TondthemeanofS2,webeginwiththeidentitynXi=1(Xi)2=nXi=1((XiX)+(X))2=nXi=1(XiX)2+nXi=1(XiX)(X)+n(X)2=nXi=1(XiX)2+n(X)2Then,ES2=E"1 nnXi=1(Xi)2(X)2#=1 nn21 n2=n1 n2:This VariationofParameters,UseofaKnownSolutiontoFindAnotherandCauchy-EulerEquationDepartmentofMathematicsIITGuwahati SU/KSK MA-102(2018) VariationofParameters,UseofaKnownSolutiontoFindAnotherandCauchy-Eule arecalledtheSupportVectorsinourproblem.withminimizingoverbandtonLagrangian(@L(t;b;a;u;v;) @b=PNi=1ui�PMi=1vi=0@L(t;b;a;u;v;) @t=�1+PNi=1ui+PMi=1vi=0yieldstheconditions1�u=1=2;1& PeterBartlett 1 Overview  AdaBoost Coordinatedescentwithotherlosses. Dualproblem:maximumentropy/I-projection. AdaBoostisiterativeprojectionmethod. Weaklylearnable,infeasible. UnnormalizedKLprojection @0=2Pni=11Yi(0+1xi)]@S @1=2Pni=11Yi(0+1xi)]xi(2.3)Whencomparedtozeroweobtainsocallednormalequations:8:Pni=1(b0+b1xi)=Pni=1YiPni=1(b0+b1xi)xi=Pni=1xiYi(2.4)Thissetofequationscanbewrittena y�b1 x(Note: y=1 nPni=1yiand x=1 nPni=1xi.)Proof:Ignoringthesecondnormalequation,startbydividingtherstnormalequationbyn:1 nnXi=1yi=b0+b1 nnXi=1xi:Rearrangingthisequation,andnotingthat y=1 nPni span(f(X))!XsuchthatkTk=1andT(f(x))=xforeveryx2X.Thus,whenviewedasmetricspacesintheisometriccategory,Banachspacesarehighlyrigid:theirlinearstructureiscompletelypreservedunderisometries,and,infact,isom 1IntroductionThislecturemainlyfocusesondimensionreduction:Johnson-LinderstraussLemmaandespeciallyitsdistributionalversion.Chi-squareddistributionisintroducedwhenthereisasumofGaussiandistributedvariabl AaronDefazioAmbiataAustralianNationalUniversity,CanberraFrancisBachINRIA-SierraProject-Team´EcoleNormaleSup´erieure,Paris,FranceSimonLacoste-JulienINRIA-SierraProject-Team´EcoleNormaleSup´erie UnderjointGaussianassumptionsweknowthatKalmansmoothingisequivalentto12ndingtheMaximumAPosterioriMAPestimatorforthestatesgiventhewholeobservationsHoweverhowdoesthisfactalgebraicallyplayoutWhatifthevari CONTENTSCONTENTSContents1Introduction12Killingtensors221IsometriesandtheLiederivative222Killingvectorsandtensors423TheLieandsynmmetricSchouten21Nijenhuisbrackets624Conservationlawsforparticles925Conse w1w218w1w2X1nXi3wiXi19w2w1w218w1w2X2nXi3wiXi192120w1w1w2E20f18w1w2X1nXi3wiXi1921w2w1w2E20f18w1w2X2nXi3wiXi19219wheretheinequalityholdsbyconvexityoffBysymmetrywehaveEf w1w2X1nXi3wiXiEf w1w2X2nXi3wiXi10