- #1

onako

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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)?

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)?

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