# Squared norms: difference or notational convenience

1. ### onako

87
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)?

Last edited: Mar 8, 2012
2. ### Office_Shredder

4,499
Staff Emeritus
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

3. ### onako

87
Thanks; I had similar reasoning. However, I'm surprised that in the literature one might find some confusing monotone transformations.