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-   -   Squared norms: difference or notational convenience (http://www.physicsforums.com/showthread.php?t=584993)

 onako Mar8-12 11:22 AM

Squared norms: difference or notational convenience

Given certain matrix $$A\in\mathbb{R}^{n\times m},$$the rank d approximation L with the same number of rows/column as A, minimizing the Frobenius norm of the difference $$||A-L||$$ 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.
$$i) ||A-K||^2$$
$$ii) \left(\frac{||A-K||}{||A||}\right)^{1/2}$$
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)?

 Office_Shredder Mar8-12 02:21 PM

Re: Squared norms: difference or notational convenience

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

 onako Mar9-12 05:35 AM

Re: Squared norms: difference or notational convenience

Thanks; I had similar reasoning. However, I'm surprised that in the literature one might find some confusing monotone transformations.

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