The optimal rank-r approximation to $\mathbf{X}$, in a least-squares sense, is given by the rank-r SVD truncation $\tilde{\mathbf{X}}$: $ \operatorname*{argmin}_{\tilde{\mathbf{X}},\,\text{s.t. rank}(\tilde{\mathbf{X}})=r} \lVert\mathbf{X}-\tilde{\mathbf{X}}\rVert_F = \tilde{\mathbf{U}}\tilde{\mathbf{\Sigma}}\tilde{\mathbf{V}}^* $