## 2 Standard Projection-Based Approach for Data Reconstruction
It is well known [44; 45] that this minimum error is given by

\[\sum_{j=1}^{n_{s}}\|\mathbf{y}_{j}-\mathbf{U}_{k}\mathbf{U}_{k}^{T}\mathbf{y}_{ j}\|_{2}^{2}=\sum_{\ell=k+1}^{r}\sigma_{\ell}^{2}, \tag{5}\]

which is the sum of the neglected singular values \(\sigma_{k+1},\ldots,\sigma_{r}\) from the SVD of \(\mathbf{Y}\). When the dimension \(n\) of snapshots is not too large, we can directly obtain the POD basis from the SVD of the snapshot matrix. However, in practice, \(n\) can be extremely large and computing the POD basis through a standard SVD might not be efficient. In this work, we use the randomized SVD to handle this problem. ## 4 Randomized SVD (rSVD)

The randomized SVD (rSVD) was proposed in [26] to reduce the computational complexity of the standard approach for computing SVD. It consists of two stages. In the first stage, random sampling is used to obtain a reduced matrix whose range approximates the range of \(\mathbf{Y}\); in the second stage, this reduced matrix is then factorized inexpensively. In particular, suppose we want to construct a low-dimensional basis with rank \(k\) by using rSVD. A random matrix \(\Omega\in\mathbb{R}^{n_{s}\times k}\) is first employed to generate the weights used in the linear combination of the data matrix \(\mathbf{Y}\) to obtain a matrix with \(k\) columns, where \(k<n_{s}\). Let

\[\mathbf{Z}=\mathbf{Y}\Omega\in\mathbb{R}^{n\times k}.\]

Then the matrix \(\mathbf{Z}\) is used to find an orthonormal basis \(\mathbf{Q}\in\mathbb{R}^{n\times k}\), which can be done through many algorithms, such as Gram-Schmidt process, Householder transformation, or the QR decomposition. At this point, we can use the basis matrix \(\mathbf{Q}\) to approximate the column span of \(\mathbf{Z}\), as well as the column span of the data matrix \(\mathbf{Y}\), i.e.,

\[\mathbf{Y}\approx\mathbf{Q}\mathbf{B},\]

where \(\mathbf{B}\in\mathbb{R}^{k\times n_{s}}\) can be obtained from the orthogonal projection of \(\mathbf{Y}\) onto the low dimensional subspace spanned by the columns of \(\mathbf{Q}\), which gives \(\mathbf{B}=\mathbf{Q}^{T}\mathbf{Y}\). The second stage of rSVD then computes the SVD of this small matrix \(\mathbf{B}\). Suppose \(\mathbf{B}=\hat{\mathbf{U}}\hat{\mathbf{Z}}\hat{\mathbf{V}}^{T}\) is the SVD of \(\mathbf{B}\), where matrices \(\hat{\mathbf{U}}=[\widehat{\mathbf{u}}_{1},\dots,\widehat{\mathbf{u}}_{p}] \in\mathbb{R}^{k\times p}\) and \(\hat{\mathbf{V}}=[\widehat{\mathbf{v}}_{1},\dots,\widehat{\mathbf{v}}_{p}] \in\mathbb{R}^{n_{s}\times p}\) are matrices with orthonormal columns, \(\hat{\Sigma}=diag(\widehat{o}_{1},\dots,\widehat{o}_{p})\in\mathbb{R}^{n \times p}\) is a diagonal matrix with \(\widehat{o}_{1}\geq\widehat{o}_{2}\geq\dots\geq\widehat{o}_{p}>0\) and \(\hat{r}=\text{rank}(\mathbf{B})\). Then, we finally obtain the approximated POD basis or the left singular vectors of \(\mathbf{Y}\) from the product \(\mathbf{Q}\hat{\mathbf{U}}\). In practice, the matrix \(\mathbf{Z}\) constructed from an \(n\)-by-\(k\) random matrix may not be a good representation of the original data matrix \(\mathbf{Y}\). Consequently, the basis matrix \(\mathbf{Q}\) may not give an accurate approximation for the column space of \(\mathbf{Y}\). To solve this issue, we consider an improved algorithm for rSVD by using an oversampling strategy given in [26], which uses an \(n\)-by-\((k+p)\) random matrix instead of \(n\)-by-\(k\), where \(p\) is a positive integer of additional random vectors. Algorithm 2 summarizes these two stages of the rSVD.