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LS solution vs. pre-averaging

  1. Jan 22, 2013 #1

    I have a system of equations [itex]\mathbf{y} = \mathbf{A}\mathbf{c}[/itex] where the entries in [itex]\mathbf{c}[/itex] are small (say, K=10 elements) and the number equations (i.e., elements in [itex]\mathbf{y}[/itex]) is huge (say, N=10000 elements).

    I want to solve now for [itex]\mathbf{c}[/itex]; this can be done using LS with the Pseudo inverse:

    [tex]\mathbf{c} = \mathbf{A}^{\dagger} \mathbf{y}[/tex]

    However, the vector [itex]\mathbf{y}[/itex] is now heavily corrupted by noise (just assume iid Gaussian).

    I could calculate the mean over M consecutive elements in [itex]\mathbf{y}[/itex] and rows in [itex]\mathbf{A}[/itex] in order to average over the noise. The system would be collapsed to a smaller system with N/M entries which would be solved via LS.

    Now I ask the question: Is this better than directly using LS with the full system?

    I doubt because that's the sense of LS. However, I was not able to "proof" this analytically.

    Any help?
  2. jcsd
  3. Jan 22, 2013 #2


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    Science Advisor

    Hey divB.

    Can you use the properties of a psuedo-inverse to show that this holds? (Recall that a pseudo-inverse has the property that C*C'*C = C)
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