# Indoor Localization Using Semi-Supervised Manifold Alignment with Dimension Expansion
## 3 Algorithm Description
### Noise Perturbation
#### 3.4.2 Impact of Noise Perturbation on Localization Accuracy  
As the value \(\Delta\) increases, we assume that the decreased values with respect to the solutions to (7), \(\mathbf{P}_{x}{}^{\mathrm{T}}\) and \(\mathbf{P}_{y}{}^{\mathrm{T}}\), are \(\mathbf{P}_{x}{}^{\mathrm{T}}\)\(-\mathbf{\Phi}_{1}\) and \(\mathbf{P}_{y}{}^{\mathrm{T}}\)\(-\mathbf{\Phi}_{2}\), respectively. We notate \(\mathbf{h}_{p}(1\leq p\leq l)\) as the estimated coordinate in \(\mathbf{H}^{\mathrm{T}}\) for the newly-collected RSS measurement \(\mathbf{r}\) without the variation \(\Delta\). For simplicity and without losing generality, we select the one-norm operation for the localization. Then, we have:  
\[||\mathbf{P}_{y}{}^{\mathrm{T}}\mathbf{h}_{p}-\mathbf{P}_{x}{}^{\mathrm{T}} \mathbf{r}^{\mathrm{T}}||_{1}=\min_{j\in\{1,2,\ldots,l\}}\{||\mathbf{P}_{y}{}^{ \mathrm{T}}\mathbf{h}_{j}-\mathbf{P}_{x}{}^{\mathrm{T}}\mathbf{r}^{\mathrm{T}} ||_{1}\} \tag{25}\]where the notation \"\(||\bullet||_{1}\)\" represents the one-norm operation.  
Under the variation \(\Delta\), we assume that the transformed vector for the newly-collected RSS measurement, \(\mathbf{r}\), is \((\mathbf{P}_{x}{}^{\mathrm{T}}\!-\!\mathbf{\Phi}_{1})\mathbf{r}^{\mathrm{T}}\), and the manifold corresponding to \(\mathbf{H}^{\mathrm{T}}\) is \(\{(\mathbf{P}_{y}{}^{\mathrm{T}}\!-\!\mathbf{\Phi}_{2})\mathbf{h}_{1}\), \((\mathbf{P}_{y}{}^{\mathrm{T}}\!-\!\mathbf{\Phi}_{2})\mathbf{h}_{2}\),..., \((\mathbf{P}_{y}{}^{\mathrm{T}}\!-\!\mathbf{\Phi}_{2})\mathbf{h}_{l}\}\). Based on this, the distance between \(\mathbf{h}_{p}\) and \(\mathbf{r}^{\mathrm{T}}\) is changed into:  
\[||(\mathbf{P}_{y}{}^{\mathrm{T}}\!-\!\mathbf{\Phi}_{2})\mathbf{h}_{p}\!-\!( \mathbf{P}_{x}{}^{\mathrm{T}}\!-\!\mathbf{\Phi}_{1})\mathbf{r}^{\mathrm{T}}|| _{1}\!=||\mathbf{P}_{y}{}^{\mathrm{T}}\mathbf{h}_{p}-\mathbf{P}_{x}{}^{ \mathrm{T}}\mathbf{r}^{\mathrm{T}}-\mathbf{\Phi}_{2}\mathbf{h}_{p}\!+\! \mathbf{\Phi}_{1}\mathbf{r}^{\mathrm{T}}||_{1}, \tag{26}\]  
and meanwhile, the distance between \(\mathbf{h}_{k}(k\in\{1,2,...,l\},k\
eq p)\) and \(\mathbf{r}^{\mathrm{T}}\) equals:  
\[||(\mathbf{P}_{y}{}^{\mathrm{T}}\!-\!\mathbf{\Phi}_{2})\mathbf{h}_{k}\!-\!( \mathbf{P}_{x}{}^{\mathrm{T}}\!-\!\mathbf{\Phi}_{1})\mathbf{r}^{\mathrm{T}}|| _{1}\!=||\mathbf{P}_{y}{}^{\mathrm{T}}\mathbf{h}_{k}\!-\!\mathbf{P}_{x}{}^{ \mathrm{T}}\mathbf{r}^{\mathrm{T}}\!-\!\mathbf{\Phi}_{2}\mathbf{h}_{k}\!+\! \mathbf{\Phi}_{1}\mathbf{r}^{\mathrm{T}}||_{1}. \tag{27}\]  
Then, it is obtained that \(||(\mathbf{P}_{y}{}^{\mathrm{T}}\!-\!\mathbf{\Phi}_{2})\mathbf{h}_{p}\!-\!( \mathbf{P}_{x}{}^{\mathrm{T}}\!-\!\mathbf{\Phi}_{1})\mathbf{r}^{\mathrm{T}}|| _{1}-||(\mathbf{P}_{y}{}^{\mathrm{T}}\!-\!\mathbf{\[PERSON]}_{2})\mathbf{h}_{k}\!- \!(\mathbf{P}_{x}{}^{\mathrm{T}}\!-\!\mathbf{\[PERSON]}_{1})\mathbf{r}^{\mathrm{T}} ||_{1}\) is equivalent to \(||\mathbf{P}_{y}{}^{\mathrm{T}}\mathbf{h}_{p}\!-\!\mathbf{P}_{x}{}^{\mathrm{T }}\mathbf{r}^{\mathrm{T}}\!-\!\mathbf{\[PERSON]}_{2}\mathbf{h}_{p}||_{1}-|| \mathbf{P}_{y}{}^{\mathrm{T}}\mathbf{h}_{k}\!-\!\mathbf{P}_{x}{}^{\mathrm{T }}\mathbf{r}^{\mathrm{T}}\!-\!\mathbf{\Phi}_{2}\mathbf{h}_{k}||_{1}\), and due to the arbitrariness of \(\mathbf{h}_{k}\), we have:  
\[||\mathbf{P}_{y}{}^{\mathrm{T}}\mathbf{h}_{p}\!-\!\mathbf{P}_{x}{}^{\mathrm{ T}}\mathbf{r}^{\mathrm{T}}\!-\!\mathbf{\Phi}_{2}\mathbf{h}_{p}||_{1}-|| \mathbf{P}_{y}{}^{\mathrm{T}}\mathbf{h}_{k}\!-\!\mathbf{P}_{x}{}^{\mathrm{T}} \mathbf{r}^{\mathrm{T}}\!-\!\mathbf{\Phi}_{2}\mathbf{h}_{k}||_{1}>0 \tag{28}\]  
Therefore, \(\mathbf{h}_{k}\) becomes the estimated location with respect to \(\mathbf{r}\), which will be deteriorated as the value \(\Delta\) increases.