## 4 The ST-LRMA Based Data Collection Method
Therefore, prior to recovering the sensing data, we could estimate the unknown measurements by some simple interpolation algorithms or the B-LRMA. From the estimated complete sensing data, we could construct the spatial correlation matrix \(\mathbf{S}\). According to our experimental results, the T-LRMA data recovery model in Equation (5) has better estimated accuracy than other algorithms. So, we use the optimized result of the T-LRMA model as the estimation of the unknown sensing data to construct \(\mathbf{S}\). Having determined the temporal and spatial constraints \(\mathbf{D}\) and \(\mathbf{S}\), we integrate the low rank approximation models in Equations (5) and (7) into a combined model to form a novel correlated sptaio-temporal constrained model, the ST-LRMA model:  
\[ \begin{split}\mathbf{X}^{*}=&\operatorname*{argmin} \limits_{\mathbf{X}}\quad\text{rank}(\mathbf{X})+\lambda_{1}\|\mathbf{X} \mathbf{D}\|_{F}^{2}+\lambda_{2}\|\mathbf{S}\mathbf{X}\|_{F}^{2},\\ &\text{subject to}\quad\mathcal{A}(\mathbf{X})=\mathbf{M}.\end{split}  \tag{12}\]  
In general, a rank minimization problem like Equation (12) is NP-hard [12]. The common practice in solving Equation (12) is to replace the rank function with the so-called matrix nuclear norm \(\|\cdot\|_{*}\) which is defined as the sum of singular values of \(\mathbf{X}\). Thus Equation (12) can be converted into a convex optimization problem [12]. Leveraging the computation and memory resource in the sink node and the data recovering delay for large scale WSNs (N nodes), we adopt the approximate algorithm in [12], which is considered to have good efficiency with a relative small memory request. This method seeks for a solution with a fixed low rank \(r\). The process can be described as follows. Suppose that the rank of targeted solution \(\mathbf{X}\) is \(r\). Then \(\mathbf{X}\) admits a skinny SVD \(\mathbf{X}=\mathbf{U}\boldsymbol{\Sigma}\mathbf{V}^{T}\), where \(\mathbf{U}\) is an \(N\times r\) unitary matrix, \(\mathbf{V}\) is a \(T\times r\) unitary matrix and \(\boldsymbol{\Sigma}\) is an \(r\times r\) diagonal matrix containing all the singular values \(\sigma_{k}\), \(k=1,2,...,r\), which are arranged in a decreasing order. We further write as \(\mathbf{X}=\mathbf{U}\boldsymbol{\Sigma}\mathbf{V}^{T}=\mathbf{L}\mathbf{R}^{T}\), where \(\mathbf{L}=\mathbf{U}\boldsymbol{\Sigma}^{1/2}\) and \(\mathbf{R}=\mathbf{V}\boldsymbol{\Sigma}^{1/2}\). So the model in Equation (12) can be replaced by the following constrained minimization model:  
\[ \begin{split}(\mathbf{L}^{*},\mathbf{R}^{*})=& \operatorname*{argmin}\limits_{\mathbf{L},\mathbf{R}}\quad\text{ rank}(\mathbf{L}\mathbf{R}^{T})+\lambda_{1}\|\mathbf{L}\mathbf{R}^{T} \mathbf{D}\|_{F}^{2}+\lambda_{2}\|\mathbf{S}\mathbf{L}\mathbf{R}^{T}\|_{F}^{2 },\\ &\text{subject to}\quad\mathcal{A}(\mathbf{L}\mathbf{R}^{T})= \mathbf{M}.\end{split}  \tag{13}\]  
The dimension of \(\mathbf{L}\) and \(\mathbf{R}\) are \(N\times r\) and \(T\times r\), respectively. In practice, we can specify an estimate to the rank \(r\) of \(\mathbf{X}\), which could be larger than the actual rank of the targeted \(\mathbf{X}\). From the lemma in [12], if the restricted isometry property holds on \(\mathcal{A}(\mathbf{X})\) and \(\text{rank}(\mathbf{X})<\text{rank}(\mathbf{L}\mathbf{R}^{T})\), then Equation (13) is equivalent to the following model:  
\[ \begin{split}(\mathbf{L}^{*},\mathbf{R}^{*})=& \operatorname*{argmin}\limits_{\mathbf{L},\mathbf{R}}\quad\|\mathbf{L}\|_{F}^ {2}+\|\mathbf{R}^{T}\|_{F}^{2}+\lambda_{1}\|\mathbf{L}\mathbf{R}^{T}\mathbf{D} \|_{F}^{2}+\lambda_{2}\|\mathbf{S}\mathbf{L}\mathbf{R}^{T}\|_{F}^{2},\\ &\text{subject to}\quad\mathcal{A}(\mathbf{L}\mathbf{R}^{T})= \mathbf{M}.\end{split}