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Rank approximation problem


More precisely, it should be called as low-rank approximation.

Wiki and Horn P449 give good discussions on it. What I want to highlight here is the following assertion in wiki:

\min_{\tilde M} \|\Sigma - U^* \tilde M V\|_F.

In order to minimize the above equation, it is said U^*\tilde{M}V should be diagonal. This is easy to prove. If U^*\tilde{M}V is not diagonal, we can always find another \tilde{M}_2 making U^*\tilde{M}_2V be the diagonal of U^*\tilde{M}V, and hence giving smaller ||*||.

But when U^*\tilde{M}V is diagonal, we can’t say \tilde{M}=U\Sigma(\tilde{M})V, right? Therefore, this proof is not rigorous.

In fact, Horn gave some theorems on this problem. See 7.4.50 and 7.4.51 on P447~448.

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