Algorithm 5:DIM-CORESET$(A,k,\epsilon )$

1:Input: A matrix $A\in {\mathbb{R}}^{n\times d}$, an integer $k\in \left[d\right]$, and an error parameter $\epsilon \in (0,1)$. 2:Output: A diagonal matrix $W\in {\mathbb{R}}^{n\times n}$ that satisfies Corollary 7. 3:${\displaystyle r:=1+\underset{i\in \left[n\right]}{max}\frac{4{∥a_i∥}^2}{{\epsilon }^4}}$ {where $a_i$ is the ith row of A} 4:$U,\Sigma ,V$: = the full SVD of $[A\mid {(r,\cdots ,r)}^T]\in {\mathbb{R}}^{n\times (d+1)}$ 5:$v_i:=\left(U_{i,1},\cdots ,U_{i,k},\frac{U_{i,k+1:d}{\Sigma }_{k+1:d,k+1:d}}{{∥{\Sigma }_{k+1:d,k+1:d}∥}_F}\right)$ for every $i\in \left[n\right]$ 6:$\widetilde{v_i}$:= the row stacking of $v_i{v}_i^T\in {\mathbb{R}}^{d\times d}$ for every $i\in \left[n\right]$ 7:$(\left\{\widetilde{v_1},\cdots ,\widetilde{v_n}\right\},u)$: = a vector summarization ${\left(\frac{\epsilon }{5k}\right)}^2$-coreset for $(\left\{\widetilde{v_1},\cdots ,\widetilde{v_n}\right\},(1,\cdots ,1))$. 8:$W:=$ a diagonal matrix in ${\mathbb{R}}^{n\times n}$, where $W_{i,i}=\sqrt{u_i}$, $\forall i\in \left[n\right]$. 9:ReturnW