Re: [eigen] LeastSquares, Pseudo Inverse & Geometry

[Doug Heisterkamp <doug@xxxxxxxxxxxxxx>]

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. 

Doug


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 ?
> :0)
>
> ++
> Thomas