## 7 RSS Difference-Aware Sparse Graph-Based Semi-Supervised Learning Method and Experimental Results  
### Sparse Graph Construction for RG-SSL Using CS Method  
Since the radio map is constructed using a step-counter-assisted RSS measurement method, the coordinate of each RP calculated by this method contains a lot of noise. In the proposed RG-SSL method, the _heat-kernel_ function is used to construct the graph and calculate the edge weights based on the Euclidean distance. However, the Euclidean distance and consequently the weights are very sensitive to noise.  
The accuracy of the generated graph will greatly affect the positioning performance. When the vertices in the graph are far away from each other, the graph weight is much smaller than the graph weight calculated for neighboring vertices. Therefore, the graph weight matrix is sparse. Since the CS method is robust to noisy data, we can use it to estimate the graph weight matrix [35]. As mentioned in Section 2, we denote the vertex set \(V=\{c_{1},c_{2},\ldots,c_{\ell},c_{\ell+1},\ldots,c_{\ell+u}\}\). Given the measurement matrix \(\mathbf{A}\) and the matrix for unknown reconstruction coefficients \(\mathbf{W}\) we can reconstruct a sparse \(\mathbf{W}\) from \(\mathbf{V}=\mathbf{A}\mathbf{W}\) using:  
\[\min_{\mathbf{W}}\|\mathbf{W}\|_{0},\;s.t.\;\mathbf{V}=\mathbf{A}\mathbf{W}, \tag{32}\]  
where \(\|\bullet\|_{0}\) denotes the \(\ell_{0}\)-norm. The \(\ell_{0}\)-norm minimization is NP-hard. However, if the solution is sparse enough, the following convex \(\ell_{1}\)-norm minimization can be used to solve the sparse representation problem:  
\[\min_{\mathbf{W}}\|\mathbf{W}\|_{1},\;s.t.\;\mathbf{V}=\mathbf{A}\mathbf{W}, \tag{33}\]  
Suppose the noise in the collected RSS is denoted by \(\xi\). Then,  
\[\mathbf{V}=\mathbf{A}\mathbf{W}+\xi=[\mathbf{A}\;\mathbf{I}]\left[\begin{array} []{c}\mathbf{W}\\ \xi\end{array}\right]=\mathbf{B}\mathbf{W}^{\prime}, \tag{34}\]  
where \(\mathbf{B}=[\mathbf{A}\;\mathbf{I}]\) and \(\mathbf{W}^{\prime}=\left[\begin{array}{c}\mathbf{W}\\ \xi\end{array}\right]\). Thus the \(\ell_{1}\)-norm minimization can be rewritten as:  
\[\min_{\mathbf{W}^{\prime}}\|\mathbf{W}^{\prime}\|_{1},\;s.t.\;\mathbf{V}= \mathbf{B}\mathbf{W}^{\prime}, \tag{35}\]  
For each \(c_{i}\) in the vertex set, the measurement matrix \(\mathbf{B}_{i}\) is constructed as \(\mathbf{B}=[c_{1},\ldots,c_{i-1},c_{i+1},\ldots,c_{\ell+u},I]\) and \(w_{i}^{\prime}\) is calculated using \(\ell_{1}\)-norm minimization:  
\[\min_{w_{i}^{\prime}}\|w_{i}^{\prime}\|_{1},\;s.t.\;c_{i}=\mathbf{B}_{i}w_{i} ^{\prime}, \tag{36}\]  
where \(w_{i}^{\prime}\) is the \(i\)-th column of the matrix \(\mathbf{W}\). Then the graph weights \(w_{ij}\) are obtained using:  
\[w_{ij}=\left\{\begin{array}{ll}w_{i}^{\prime}(j),&\text{if }j<i\\ w_{i}^{\prime}(j-1),&\text{if }j>i\\ 0,&\text{if }j=i\end{array}\right., \tag{37}\]  
\begin{table}
\begin{tabular}{l c c c} \hline
**Algorithm** & **Cumulative Probability (Location Error Is 2 m)** & **Average Error (m)** & **Maximum Error (m)** \\ \hline RG-SSL & 63.5\% & 2.07 & 4 \\ G-SSL & 60\% & 2.13 & 4.5 \\ SPORT & 60\% & 2.15 & 4.5 \\ SCTW & 54.3\% & 2.24 & 5.5 \\ Original data & 42.9\% & 2.89 & 10 \\ \hline \end{tabular}
\end{table}
Table 1: Comparison of different algorithms.  
where \(i,j\in\{1,2,\ldots,\ell+u\}\) and \(w^{\prime}_{i}(j)\) denotes the \(j\)-th element of vector \(w^{\prime}\).