Talk:Levenberg–Marquardt algorithm: Difference between revisions

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== L-M is for nonlinear least squares problems only ==

The article needs to make clear that the L-M algorithm is for solving nonlinear least squares problems, not the more general class of nonlinear optimization problems. In fact, the introduction used to give the impression that the L-M algorithm is used to solve general nonlinear optimization problems, which is not true.

== Functions vs. Values ==

The article talks about "the functions <math>f(x_i,\boldsymbol \beta+\boldsymbol \delta)</math>". This confused the heck out of me, until I realized that what is meant is the values of the function <math>f(x,\boldsymbol \beta+\boldsymbol \delta)</math> at <math>x_i</math>. Similar confused terminology is used at other points in the article, for example when calling <math>J_i</math> the gradient. These kind of terminology issues make the article hard to read. <span style="font-size: smaller;" class="autosigned">— Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[Special:Contributions/80.65.109.181|80.65.109.181]] ([[User talk:80.65.109.181|talk]]) 12:09, 13 September 2013 (UTC)</span><!-- Template:Unsigned IP --> <!--Autosigned by SineBot-->

== Inconsistency ==

In the article is says that Levenberg's contribution is in introducing lambda. But at the same time in the references it says that he was the first to publish the algorithm in 1944 or something. This seems a bit contradictory to me.

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:Both formulas have the same meaning, namely '''J'''^T'''Jq''' + &lambda;'''q''' = &minus;'''J'''^T'''f'''. As far as I am concerned, it is a matter of taste which you prefer, but the second is perhaps clearer. -- [[User:Jitse Niesen|Jitse Niesen]] ([[User talk:Jitse Niesen|talk]]) 20:46, 30 August 2005 (UTC)

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All formulas have a lack of negative sign. For example:

:<math>\mathbf{(J^{T}J)\boldsymbol \delta = J^{T} [y - f(\boldsymbol \beta)]} \!</math>

should be:

:<math>\mathbf{(J^{T}J)\boldsymbol \delta = -J^{T} [y - f(\boldsymbol \beta)]} \!</math>

finding this bug took quite some time. And it should be pointed out, how to get to this formula. --vogt31337 9:00 26 January 2012 (UTC) <small><span class="autosigned">— Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[User:Vogt31337|Vogt31337]] ([[User talk:Vogt31337|talk]] • [[Special:Contributions/Vogt31337|contribs]]) </span></small><!-- Template:Unsigned --> <!--Autosigned by SineBot-->

@ Vogt31337:

i think there is no sign missing since '''J''' is the jacobian of '''f''' and '''f''' has a negativ sign in '''S'''. the goal is to minimize '''S''' and therefore we want to go in direction of negativ jacobian of

'''S''' which turns out to be the negative jacobian of '''-f''' and thus '''-(-J) = J'''. -- Georg

@ Vogt31337: I agree with you. -- D <!-- Template:Unsigned IP --><small class="autosigned">—&nbsp;Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[Special:Contributions/192.55.54.42|192.55.54.42]] ([[User talk:192.55.54.42#top|talk]]) 23:13, 28 June 2019 (UTC)</small> <!--Autosigned by SineBot-->

==Transpose==

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Any answers/thoughts/corrections would be appreciated. Thanks. [[User:BenFrantzDale|—Ben FrantzDale]] ([[User talk:BenFrantzDale|talk]]) 22:04, 7 December 2008 (UTC)

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In between the section Solution there is the sentence:

: >> Note that the gradient of ''S'' with respect to ''δ'' equals <math>-2(\mathbf{J}^{T} [\mathbf{y} - \mathbf{f}(\boldsymbol \beta) ] )^T </math> <<

: Is it not supposed to be: ... gradient of ''S'' with respect to '''&beta;''' ?? -Georg <span style="font-size: smaller;" class="autosigned">— Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[Special:Contributions/31.18.248.89|31.18.248.89]] ([[User talk:31.18.248.89|talk]]) 20:45, 27 May 2014 (UTC)</span><!-- Template:Unsigned IP --> <!--Autosigned by SineBot-->

== Notes and references==

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http://www.siam.org/books/textbooks/fr18_book.pdf

We're trying to minimize the error, so taking the Jacobian of the original function wouldn't make sense. <span style="font-size: smaller;" class="autosigned">— Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[Special:Contributions/129.132.154.39|129.132.154.39]] ([[User talk:129.132.154.39|talk]]) 10:42, 20 October 2011 (UTC)</span><!-- Template:Unsigned IP --> <!--Autosigned by SineBot-->

== Covariance of the solution ==

There should be discussion of the covariance of the solution. I think it's that if J is left-multiplied by the inverse of the input covariance, than <math>J^\top \Sigma^{-1} J</math> is the covariance of the output, or somesuch. [[User:BenFrantzDale|—Ben FrantzDale]] ([[User talk:BenFrantzDale|talk]]) 17:20, 26 October 2011 (UTC)

== External links modified ==

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== Broken images ==

Hi just a note to say that the image links on this page appear to be broken. Thanks. <!-- Template:Unsigned IP --><small class="autosigned">—&nbsp;Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[Special:Contributions/2A01:110:8012:1012:0:0:0:82|2A01:110:8012:1012:0:0:0:82]] ([[User talk:2A01:110:8012:1012:0:0:0:82#top|talk]]) 11:59, 17 January 2018 (UTC)</small> <!--Autosigned by SineBot-->