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The Levenberg-Marquardt Algorithm

TheLevenberg-MarquardtAlgorithmAnanthRan ganathan8thJune20041 IntroductionTheLevenberg-Marquardt(LM) rstshowntobea , anotherperspective onthealgorithmis providedbyconsideringit asa solutionis calledNonlinearLeastSquaresMinimization. Thisimpliesthatthefunctiontobeminimizedi s ofthefollowingspecialform:f x 12m j 1r2j x wherex x1 x2 xn is a vector, andeachrjis a functionfrom nto . Therjarereferredtoasaresidualsandit is assumedthatm make matterseasier,fis representedasaresidualvectorr: n mde nedbyr x r1 x r2 x rm x Now,fcanberewrittenasf x 12 r x 2. nedasJ x rj xi, 1 j m, 1 i rstconsiderthelinearcasewhereeveryrifunc tionislinear. Here,theJacobianisconstantandwecanrepres entrasa hyperplanethroughspace,sothatfis givenbythequadraticf x 12 Jx r 0 2. We alsoget f x JT Jx r and 2f x JTJ. Solvingforthemin-imumbysetting f x 0 , weobtainxmin JTJ 1 JTr, whichis ,non-linearcase,wehave f x m j 1rj x rj x J x Tr x (1) 2f x J x TJ x m j 1rj x 2rj x (2)Thedistinctive propertyofleast-squaresproblemsisthatgiv entheJacobianmatrixJ, wecanessentiallygettheHessian( 2f x ) forfreeifit ispossibletoapproximatetherjs bylinearfunctions( 2rj x aresmall)ortheresiduals(rj x ) 2f x J x TJ x (3)which

algorithm is rst shown to be a blend of vanilla gradient descent and Gauss-Newton iteration. Subsequently, another perspective on the algorithm is provided by considering it as a trust-region method. 2 The Problem The problem for which the LM algorithm provides a solution is called Nonlinear Least Squares Minimization. This implies that the ...

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