低秩逼近证明

低秩逼近证明， 发现以前搞复杂了

\[\begin{align} \min_{r(X)\leq r}\|A - X\| &= \min_{U^TU=V^TV =I, r(\Sigma)=r} \| D - U \Sigma V^T\| \\ & \leq \min_{U^TU=V^TV =I, r(\Sigma)=r } \| D - U D V^T\| + \| U D V^T - U \Sigma V^T\| \\ & = \min_{U^TU=V^TV =I} \| D - U D V^T\| + \min_{r(\Sigma) = r } \| D - \Sigma \| \\ & = \min_{ r(\Sigma) = r } \| D - \Sigma \| \end{align} \]