单纯形上多项式核最小二乘正则化算法的逼近阶

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1、Chap1IntroductionLet(X，d)beacompactmetricspaceandletY=R．LetPbeaprobabilitydistribu-tiononZ：=X×Y．111eleastsquare明．or(orgeneralizationerror)([1][2])forafunctionf：X_yisdefinedas，-．￡(，)=／(，(z)一可)2dp(1．1)Jz111efunctionthatminimizestheerroriscalledtheregressionfunction．Itisgivenby

2、，．厶(z)=／ydp(ylx)，z∈x(1．2)．，ywhereJD(·lz)istheconditionalprobabilitymeasureatzinducedby』D．Thetargetoftheregressionproblemistolearntheregressionfunctionortofindgoodapproximationsfromrandomsamples．Theleast-squareregularizedalgorithmfortheregressionproblemisadiscreteleast-square

3、problemassociatedwithaMercerkernel([3])．LetK：X×X_ybecontinuous，symmetric，andpositivesemi—definite,i．e．foranyfinitesetofdistinctpointsXl，．．．，zf)cx，thematrix(K(xi，巧))：，J：1ispositivesemi．definite．SuchafunctioniscalledaMercerkernel．TheReproducingKernelHilbertSpace(RKHS)冗Kassocia

4、tedwiththekernelK(【3】)isdefinedtobetheclosureofthelinearspanofthesetoffunctions{％：=g(x，·)：z∈x)．withtheinnerproduct<·，·>HK---<·，·>Ksatisfying<恐，％>K=g(x，可)．Thatis，<∑iQi玩t，邑岛‰>K=∑tJaifljK(xi，协)．Thereproducingpropertytakestheform<蠡0，f>K=．厂(z)，Vz∈X，f∈7ZK．Denotec(x)asthespaceofcon

5、tinuousfunctionsonXwiththenorm”‰．LetK=supz∈x护币习．ThentheabovereproducingpropertytellsUSthatfll。。≤KIIt厂IIK，vf∈咒K．(1．3)Theleast-squareregularizedalgorithmfortheregressionproblemassociatedwiththeMercerkernelKisdefinedtobetheminimizerofthefollowingleast-squareopti·mizationproblem

6、involvingasetofrandomsamplesz：={乞)罂1={(婉，犰))罂l∈扩independentlydrawnaccordingtop：左=，z，A：=argfmETtlnK{去m(mt)一鼽)2+Allfllff()(1．4)∑：l浙江大学硕士学位论文Chap1Introduction!Im!!!!!!=!!!!!!!=!!!=!!!!!!!E!!!!!!!!!!!!!!==!!!!==!=!!!!!==!!!!!!!!!!!!!!!!!!!!!!=!s!HereA≥0isaconstantcalledtheregula

7、rizationparameter．Usuallyitischosentodependonm：A=A(m)，andliram。ooA(m)=0．Convergenceratesofregularizationalgorithmsfortheregressionforclassificationinlearningtheoryhavebeenstudiedrecentlybymanymathematician．([1112114][5])Throughoutthispaper,weassumethatforsomeM≥0，p(。Ix)isalmo

8、steverywheresupportedon[-M，M]，thatis，IYI≤Malmostsurely(withrespectto力．Itfol