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I am new to optimization theory. I have a following question. For a given $X = [x_1 x_2 \ldots x_N] \in \mathbb{C}^{N \times N}$, where $x_i \in \mathbb{C}^{N\times 1}$ for $i \in \{1,\ldots,N\}$, $U = [u_1 u_2 \ldots u_N]\in \mathbb{C}^{N \times N}$, is there any possibility of obtaining a closed form solution of

$$\min_{U \in \mathbb{C}^{N\times N}}\{\|U-X\|_F^2: \|u_i\|_2^2=1\}$$

where $\|\cdot\|_2$ and $\|\cdot\|_F$ denotes the Euclidean and Frobenius norm respectively. I know how to find the solution for $\|U\|_2^2=1$, however, I do not see how to solve it when we have $\|u_i\|^2=1$.

I may be overthinking. I can just take the $X$ and normalize each column. I am not sure that would be the right approach.

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  • $\begingroup$ We have $\|U-X\|_F^2=\sum_i \|u_i-x_i\|_2^2$ and your approach would be correct. $\endgroup$ – Markus Sprecher Apr 12 at 5:02
  • $\begingroup$ Great! Thanks so much! $\endgroup$ – OptimusPrime Apr 12 at 5:42

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