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Homogeneous least squares

  1. May 18, 2012 #1
    Given a homogeneous linear least squares problem:
    [tex]
    A^{T}y=0
    [/tex]

    What is the difference between minimizing
    [tex]
    y^{T}AA^{T}y
    [/tex] (the least square error)

    and:

    [tex]
    y^{T}AA^{+}y=y^{T}A(A^{T}A)^{-1}A^{T}y
    [/tex]

    ?

    Thanks.
     
  2. jcsd
  3. May 18, 2012 #2

    chiro

    User Avatar
    Science Advisor

    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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