Algorithm 1 . Procedure to obtain a low-rank approximation of the data matrix X.

1: procedure LowRankApproximation ( $\mathbf{X}$, ξ)

2:    Returns the dimension of the data matrix $\mathbf{X}$

3:     $n,d\leftarrow$ DIMENSION $(\mathbf{X})$ $\text{⊳}n$: #row, d: #feature

4:    Utilizing either Truncated SVD or Randomized SVD

5:     $\mathbf{U},\mathbf{S},{\mathbf{V}}^{\mathrm{\top }}\leftarrow$ SVD(∗) $(\mathbf{X})$ $\text{⊳}n$ see “SVD with Row Reduction (SVD-RR) & Randomized SVD”

6:    Computes the rank of matrix $\mathbf{X}$ using eigenvalues

7:     $r\leftarrow$ RANK $(\mathbf{S},\xi )$ $\text{⊳}n$ see “Determining the Rank”

8:    The low-rank components of the matrices are extracted

9:     $\hat{\mathbf{U}}\leftarrow \mathbf{U}[:,1:r]$

10:    $\hat{\mathbf{S}}\leftarrow \mathbf{S}[1:r,1:r]$

11:    $\hat{\mathbf{V}}\leftarrow \mathbf{V}[1:r,1:r]$

12:   return $n,d,r,\hat{\mathbf{U}},\hat{\mathbf{S}},\hat{\mathbf{V}}$

13: end procedure