Homogeneous least squares

[azay]

Given a homogeneous linear least squares problem:
$$A^{T}y=0$$

What is the difference between minimizing
$$y^{T}AA^{T}y$$ (the least square error)

and:

$$y^{T}AA^{+}y=y^{T}A(A^{T}A)^{-1}A^{T}y$$

?

Thanks.

 

[chiro]

Given a homogeneous linear least squares problem:
$$A^{T}y=0$$

What is the difference between minimizing
$$y^{T}AA^{T}y$$ (the least square error)

and:

$$y^{T}AA^{+}y=y^{T}A(A^{T}A)^{-1}A^{T}y$$

?

Thanks.

Hey azay and welcome to the forums.

The difference has to do with how X is decomposed. The pseudo-inverse has the 'properties' that you would expect for an inverse but it's not the same.

According to this:

http://en.wikipedia.org/wiki/Linear_least_squares_(mathematics)#Computation

The first uses a QR decomposition, and the second uses properties related to the pseudo-inverse from a Singular Value Decomposition (SVD).

I am not exactly sure of the deep details myself, but I'm sure you can use the above link to answer more specific questions.