Talk:Levenberg–Marquardt algorithm: Difference between revisions

Revision as of 22:29, 4 February 2008

Revision as of 03:11, 21 February 2008

Line 44:

I believe Marquardt's suggested improvement was actually to scale the approximate hessian matrix, <math>J^{T}J</math>. The scaling had an effect similar to replacing the identity matrix with the diagonal of the hessian approximation. Marquardt suggests scaling the hessian approximation by an amount that makes the diagonal elements ones. Adding a constant diagonal matrix to the scaled matrix is similar to adding a proportion of the diagonal elements to the unscaled matrix. The scaling applied to the other elements of the hessian approximation improves the condition of the matrix. I suggest the following:

<math>q = \Sigma_JSigma_{JJ}[\hat{J}^T\hat{J} + \lambda{}I]^{-1}\hat{J}^T [y - f(p)]</math>

where <math>\Sigma_JSigma_{JJ}</math> is the square diagonal matrix:

<math>\hatSigma_{JJJ} = J\Sigma_J^T\Sigma_J</math>

with

<math>\Sigma_J = [\mbox{diag}[(J^TJ])]^{-\frac{1}{2}}</math>

and the scaled Jacobian, <math>\hat{J}</math>, is:

<math>\hat{J} = \Sigma_{JJ}J</math>

<math>\hat{J} = J\Sigma_J</math>

The square matrix, <math>\hat{J}^T\hat{J}</math>, is then a scaled version