Homogeneous least squares

azay
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Given a homogeneous linear least squares problem:
<br /> A^{T}y=0<br />

What is the difference between minimizing
<br /> y^{T}AA^{T}y<br /> (the least square error)

and:

<br /> y^{T}AA^{+}y=y^{T}A(A^{T}A)^{-1}A^{T}y<br />

?

Thanks.
 
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azay said:
Given a homogeneous linear least squares problem:
<br /> A^{T}y=0<br />

What is the difference between minimizing
<br /> y^{T}AA^{T}y<br /> (the least square error)

and:

<br /> y^{T}AA^{+}y=y^{T}A(A^{T}A)^{-1}A^{T}y<br />

?

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.
 

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