## 2 Standard Projection-Based Approach for Data Reconstruction  
This section describes how SVD is used to reconstruct missing components of incomplete data. We will approximate an incomplete data vector by projecting on a subspace spanned by a basis that represents other related complete data vectors. Let \(\{\mathbf{y_{1}},\mathbf{y_{2}},\ldots,\mathbf{y_{n_{s}}}\}\) be a complete data set. Define \(\mathcal{Y}:=\{\mathbf{y_{1}},\mathbf{y_{2}},\ldots,\mathbf{y_{n_{s}}}\}\subset \mathbb{R}^{n}\) and \(\mathbf{Y}=[\mathbf{y_{1}},\mathbf{y_{2}},\ldots,\mathbf{y_{n_{s}}}]\in \mathbb{R}^{n\times n_{s}}\). Suppose \(\widehat{\mathbf{y}}\) is an incomplete sample that has \(n_{c}\) known components and \(n_{g}=n-n_{c}\) unknown components. Let \(\mathcal{C}:=\{c_{1},c_{2},\ldots,c_{n_{c}}\}\subset\{1,2,\ldots,n\}\) be the indices of the _known_ components in \(\widehat{\mathbf{y}}\) and define \(\mathbf{C}=[\mathbf{e}_{c_{1}},\ldots,\mathbf{e}_{c_{n_{c}}}]\in\mathbb{R}^{ n\times n_{c}}\), where \(\mathbf{e}_{c_{i}}=[0,\ldots,0,1,0,\ldots,0]^{T}\in\mathbb{R}^{n}\) is the \(c_{i}\)-th column of the identity matrix \(\mathbf{I}_{n}\in\mathbb{R}^{n\times n}\), for \(i=1,\ldots,m\). Note that pre-multiplying \(\mathbf{C}^{T}\) is equivalent to extracting the \(n_{c}\) rows corresponding to the indices \(c_{1},\ldots,c_{n_{c}}\). Similarly, let \(\mathcal{G}=\{g_{1},g_{2},\ldots,g_{n_{g}}\}\subset\{1,2,\ldots,n\}\) be the indices of the _unknown_ components in \(\widehat{\mathbf{y}}\) and define \(\mathbf{G}=[\mathbf{e}_{g_{1}},\ldots,\mathbf{e}_{g_{n_{g}}}]\in\mathbb{R}^{ n\times n_{g}}\). Let \(\widehat{\mathbf{y}}_{c}:=\mathbf{C}^{T}\widehat{\mathbf{y}}\in\mathbb{R}^{n_{c}}\) and \(\widehat{\mathbf{y}}_{g}:=\mathbf{G}^{T}\widehat{\mathbf{y}}\in\mathbb{R}^{n_ {g}}\). Then, the known components and the unknown components are contained in the vectors \(\widehat{\mathbf{y}}_{c}\) and \(\widehat{\mathbf{y}}_{g}\), respectively. The missing components contained in \(\widehat{\mathbf{y}}_{g}\) will be approximated by first projecting \(\widehat{\mathbf{y}}\) onto the column span of a basis matrix \(\mathbf{V}\) with rank \(k\), where \(k\leq r,r:=\text{rank}(\mathbf{Y})\). i.e.,  
\[\widehat{\mathbf{y}}\;\approx\;\mathbf{V}\mathbf{a},\qquad\text{or}\qquad \widehat{\mathbf{y}}_{c}\approx\mathbf{V}_{c}\mathbf{a}\;\text{and}\;\;\widehat {\mathbf{y}}_{g}\approx\mathbf{V}_{g}\mathbf{a},\]  
for some coefficient vector \(\mathbf{a}\in\mathbb{R}^{k}\), and where \(\mathbf{V}_{c}:=\mathbf{C}^{T}\mathbf{V}\in\mathbb{R}^{n_{c}\times k}\), \(\mathbf{V}_{g}:=\mathbf{G}^{T}\mathbf{V}\in\mathbb{R}^{n_{g}\times k}\). The known components contained in \(\widehat{\mathbf{y}}_{c}\) are then used to determine the coefficient vector \(\mathbf{a}\) through the approximation \(\widehat{\mathbf{y}}_{c}\approx\mathbf{V}_{c}\mathbf{a}\) from the following least-squares problem:  
\[\min_{\mathbf{a}\in\mathbb{R}^{k}}\|\widehat{\mathbf{y}}_{c}-\mathbf{V}_{c} \mathbf{a}\|_{2}^{2}. \tag{1}\]  
The solution obtained from the corresponding normal equation of the above problem is given by \(\mathbf{a}=\mathbf{V}_{c}^{+}\widehat{\mathbf{y}}_{c}\), where \(\mathbf{V}_{c}^{+}=(\mathbf{V}_{c}^{T}\mathbf{V}_{c})^{-1}\mathbf{V}_{c}^{T}\) is the Moore-Penrose pseudo-inverse. That is,  
\[\widehat{\mathbf{y}}_{g}\;\approx\;\mathbf{V}_{g}\mathbf{a}\;=\;\mathbf{V}_{g }\mathbf{V}_{c}^{+}\widehat{\mathbf{y}}_{c}. \tag{2}\]  
The steps described above, which is called the POD-LS approach or standard LS approach, are summarized in Algorithm 1. ```
1:Input: \(\mathbf{y}\), \(\mathbf{y}\), \(\mathbf{y}\),


```
1:
2:- Complete snapshot set \(\{\mathbf{y}_{j}\}_{j=1}^{n_{s}}\subset\mathbb{R}^{n}\) and, dimension \(k\leq rank(\{\mathbf{y}_{j}\}_{j=1}^{n_{c}})\)
3:- Incomplete data \(\widehat{\mathbf{y}}\in\mathbb{R}^{n}\) with known entries in \(\widehat{\mathbf{y}}_{c}\in\mathbb{R}^{n_{c}}\) and unknown entries in \(\widehat{\mathbf{y}}_{g}\in\mathbb{R}^{n_{g}}\), where \(n_{c}+n_{g}=n\).