Squared norms: difference or notational convenience

  1. Given certain matrix [tex]A\in\mathbb{R}^{n\times m},[/tex]the rank d approximation L with the same number of rows/column as A, minimizing the Frobenius norm of the difference [tex]||A-L||[/tex] is matrix obtained by singular value decomposition of A, with only d dominant singular values (the rest is simply set to zero).

    However, I often encounter the minimization of the adapted norm, such as various kinds of normalization on the norm, ie.
    [tex]i) ||A-K||^2[/tex]
    [tex]ii) \left(\frac{||A-K||}{||A||}\right)^{1/2}[/tex]
    and I'm not sure if the solution L from the above non-squared Frobenius norm coincides with the normalized Frobenius norm solution from i) and ii).
    Isn't it the case that K should be L, but appropriately scaled for i) and/or ii)?
     
    Last edited: Mar 8, 2012
  2. jcsd
  3. Office_Shredder

    Office_Shredder 4,500
    Staff Emeritus
    Science Advisor
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    Essentially you're given a function f(K) and asked to minimize it. You're then asked to minimize f(K)^2 and f(K)/constant. All of these functions have the same minimum because the operations you are applying to f are all monotone
     
  4. Thanks; I had similar reasoning. However, I'm surprised that in the literature one might find some confusing monotone transformations.
     
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