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:
In order to minimize the above equation, it is said should be diagonal. This is easy to prove. If is not diagonal, we can always find another making be the diagonal of , and hence giving smaller ||*||.
But when is diagonal, we can’t say , 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.