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Math Help - Element-wise vector inversion

  1. #1
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    Element-wise vector inversion

    Hello,

    I have a problem of needing to perform an operation on a vector which results in each element being inverted. i.e.

    f(x) = y, where

    x = [x1 x2 ... xN] and y = [1/x1 1/x2 ... 1/xN]

    I'd really appreciate all of your help!

    Fran
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  2. #2
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    Quote Originally Posted by fp04 View Post
    Hello,

    I have a problem of needing to perform an operation on a vector which results in each element being inverted. i.e.

    f(x) = y, where

    x = [x1 x2 ... xN] and y = [1/x1 1/x2 ... 1/xN]

    I'd really appreciate all of your help!

    Fran
    I don't get it: what is the problem? Of course, you can apply your function f ONLY on vectors whose entries are all \neq 0 , but for that I can't see any problem

    Tonio
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  3. #3
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    Sorry, maybe I wasn't clear. Let me give you the entire problem I'm solving....

    I work in the field of MRI Physics. I need to perform the following optimisation:

    min{m1,m2} l2norm(Fm1 - y1) + l2norm(Fm2 - y2) + lambda* l1norm(f(m1,m2))

    F is the Fourier matrix, m1 and m2 are vectors of length of 512^2, y1 and y2 are vectors of the same length and lambda is a normalisation constant.

    The function f divides vectors m1 and m2 in an elementwise manner.

    As the vectors are so large I would like to use a conjugate gradient method to perform the optimisation. Therefore, I need to take the derivative of the function w.r.t m1 and m2. However, the elementwise division of m1 and m2 is undifferentiable.

    So, I need to rewrite the elementwise division of two vectors in a way which is differentiable. Therein lies the problem!

    Is this clear...?
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  4. #4
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    Quote Originally Posted by fp04 View Post
    Sorry, maybe I wasn't clear. Let me give you the entire problem I'm solving....

    I work in the field of MRI Physics. I need to perform the following optimisation:

    min{m1,m2} l2norm(Fm1 - y1) + l2norm(Fm2 - y2) + lambda* l1norm(f(m1,m2))

    F is the Fourier matrix, m1 and m2 are vectors of length of 512^2, y1 and y2 are vectors of the same length and lambda is a normalisation constant.

    The function f divides vectors m1 and m2 in an elementwise manner.

    As the vectors are so large I would like to use a conjugate gradient method to perform the optimisation. Therefore, I need to take the derivative of the function w.r.t m1 and m2. However, the elementwise division of m1 and m2 is undifferentiable.

    So, I need to rewrite the elementwise division of two vectors in a way which is differentiable. Therein lies the problem!

    Is this clear...?

    Not to me but perhaps someone else knows this stuff.

    Tonio
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