|Re: [eigen] LeastSquares, Pseudo Inverse & Geometry|
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I think what the method is doing is a least squares with a constraint. That
is, solving Ax=0 subject to ||x|| =1. It does this by minimizing ||Ax||^2
with constraint ||x||^2 =1. You would replace the constraint with a Lagrange
multiplier, set the first derivative of the objective function to zero, and
reach an eigenvalue equation of A^T A, with the eigenvector corresponding to
the smallest eigenvalue as the solution.
I don't know the proper name for the method, but I have used it quite a bit
(but not in eigen, as I am new to eigen). Though I would normally use svd on
A to get the eigenvector of A^T A.
On Monday 25 January 2010 04:52:57 am Thomas Capricelli wrote:
> That could be a difference, but i cant find information about using
> eigenvalue problems to fit hyperplanes....
> On wikipedia, none of total/ordinary LS page mentions eigenvalue methods,
> but http://en.wikipedia.org/wiki/Simple_linear_regression at least says
> "The fitted line has the slope equal to the correlation between y and x
> corrected by the ratio of standard deviations of these variables. The
> intercept of the fitted line is such that it passes through the center of
> mass (x, y) of the data points.", which looks very similar to what our
> current "LeastSquares" does.
> So this would be the same as what you call OLS and what the general least
> square i mentionned would do... ?
> Benoit, according to the log, you're the one who did that.... any thought ?