# Indoor Localization Using Semi-Supervised Manifold Alignment with Dimension Expansion
## 3 Algorithm Description
### Noise Perturbation  
Considering the noise perturbation in the indoor environment, we focus on the impact of an increment of \(\mathbf{G}\), \(\mathbf{\Delta}\), on the localization performance in this section. In this case, the objective function in (6) can be converted into:  
\[ \begin{split}&\min_{\mathbf{r}_{y}}\left(\text{Tr}(\mathbf{P}_{x} ^{\text{T}}\mathbf{H}\mathbf{H}^{\text{T}}\mathbf{P}_{x}+\mathbf{P}_{y}^{ \text{T}}(\mathbf{G}+\mathbf{\Delta})(\mathbf{G}+\mathbf{\Delta})^{\text{T}} \mathbf{P}_{y}-\mathbf{P}_{x}^{\text{T}}\mathbf{H}(\mathbf{G}+\mathbf{\Delta})^ {\text{T}}\mathbf{P}_{y}-\mathbf{P}_{y}^{\text{T}}(\mathbf{G}+\mathbf{\Delta}) \mathbf{H}^{\text{T}}\mathbf{P}_{x})+\right.\\ &\qquad\left.2\text{Tr}(\mathbf{P}_{x}^{\text{T}}\mathbf{H} \mathbf{L}_{y}^{\text{T}}\mathbf{P}_{x})+2\text{Tr}(\mathbf{P}_{y}^{\text{T}} (\mathbf{G}+\mathbf{\Delta})\mathbf{L}_{y}(\mathbf{G}+\mathbf{\Delta})^{ \text{T}}\mathbf{P}_{y})\right\}\\ &=\min_{\mathbf{P}_{x}\mathbf{P}_{y}}\left\{\text{Tr}(\mathbf{P} _{x}^{\text{T}}\mathbf{H}\mathbf{H}^{\text{T}}\mathbf{P}_{x}+\mathbf{P}_{y}^{ \text{T}}\mathbf{G}\mathbf{G}^{\text{T}}\mathbf{P}_{y}-\mathbf{P}_{x}^{\text{ T}}\mathbf{H}\mathbf{G}^{\text{T}}\mathbf{P}_{y}-\mathbf{P}_{y}^{\text{T}} \mathbf{G}\mathbf{H}^{\text{T}}\mathbf{P}_{x})+\text{Tr}(\mathbf{P}_{y}^{\text {T}}\mathbf{G}\mathbf{\Delta}^{\text{T}}\mathbf{P}_{y}+\mathbf{P}_{y}^{\text{ T}}\mathbf{\Delta}^{\text{T}}\mathbf{G}\mathbf{P}_{y}+\right.\\ &\qquad\left.\mathbf{P}_{y}^{\text{T}}\mathbf{\Delta}\mathbf{P}_{ y}-\mathbf{P}_{x}^{\text{T}}\mathbf{H}\mathbf{\Delta}^{\text{T}}\mathbf{P}_{y}- \mathbf{P}_{y}^{\text{T}}\mathbf{\Delta}\mathbf{H}^{\text{T}}\mathbf{P}_{x})+ 2\text{Tr}(\mathbf{P}_{y}^{\text{T}}\mathbf{H}\mathbf{L}_{x}\mathbf{H}^{ \text{T}}\mathbf{P}_{x})+2\text{Tr}(\mathbf{P}_{y}^{\text{T}}\mathbf{G} \mathbf{L}_{y}\mathbf{G}^{\text{T}}\mathbf{P}_{y})+\\ &\qquad\left.2\text{Tr}(\mathbf{P}_{y}^{\text{T}}\mathbf{G} \mathbf{L}_{y}\mathbf{\Delta}^{\text{T}}\mathbf{P}_{y})+2\text{Tr}(\mathbf{P} _{y}^{\text{T}}\mathbf{\Delta}\mathbf{L}_{y}\mathbf{\Delta}^{\text{T}}\mathbf{ P}_{y})+2\text{Tr}(\mathbf{P}_{y}^{\text{T}}\mathbf{\Delta}\mathbf{L}_{y} \mathbf{\Delta}^{\text{T}}\mathbf{P}_{y})\right\}\end{split}  \tag{15}\]  
Similarly, we rewrite (15) as:  
\[ \begin{split}&\min_{\mathbf{P}}\left\{\text{Tr}(\mathbf{P}^{ \text{T}}\mathbf{Z}\mathbf{\Omega}\mathbf{Z}^{\text{T}}\mathbf{P}+\mathbf{P}^ {\text{T}}\mathbf{Z}\mathbf{\Omega}\left(\begin{array}{cc}0&0\\ 0&\mathbf{\Delta}\end{array}\right)^{\text{T}}\mathbf{P}+\mathbf{P}^{\text{T}} \left(\begin{array}{cc}0&0\\ 0&\mathbf{\Delta}\end{array}\right)\mathbf{\Omega}\mathbf{Z}^{\text{T}}\mathbf{P}+ \right.\\ &\left.\mathbf{P}^{\text{T}}\left(\begin{array}{cc}0&0\\ 0&\mathbf{\Delta}\end{array}\right)\mathbf{\Omega}\left(\begin{array}{cc}0&0\\ 0&\mathbf{\Delta}\end{array}\right)^{\text{T}}\mathbf{P})\right\}&\text{s.t.} \mathbf{P}^{\text{T}}\mathbf{Z}\mathbf{Z}^{\text{T}}\mathbf{P}=\mathbf{I}^{ \prime},\mathbf{P}^{\text{T}}\mathbf{Z}\mathbf{e}=\mathbf{0}\end{split}  \tag{16}\]  
To obtain the solution of \(\mathbf{P}\) in (16), we construct the Lagrange equation as follows. \[ \begin{split}& L(\mathbf{P})=\mathbf{P}^{\text{T}}\mathbf{Z} \mathbf{\Omega}\mathbf{Z}^{\text{T}}\mathbf{P}+\mathbf{P}^{\text{T}}\mathbf{Z} \mathbf{\Omega}\left(\begin{array}{cc}0&0\\ 0&\mathbf{\Delta}\end{array}\right)^{\text{T}}\mathbf{P}+\mathbf{P}^{\text{T}} \left(\begin{array}{cc}0&0\\ 0&\mathbf{\Delta}\end{array}\right)\mathbf{\Omega}\mathbf{Z}^{\text{T}}\mathbf{P}+ \\ &\mathbf{P}^{\text{T}}\left(\begin{array}{cc}0&0\\ 0&\mathbf{\Delta}\end{array}\right)\mathbf{\Omega}\left(\begin{array}{cc}0&0\\ 0&\mathbf{\Delta}\end{array}\right)^{\text{T}}\mathbf{P}-\lambda(\mathbf{P}^{ \text{T}}\mathbf{Z}\mathbf{Z}^{\text{T}}\mathbf{P}-\mathbf{I}^{\prime})-\mu \mathbf{P}^{\text{T}}\mathbf{Z}\mathbf{e}\end{split}  \tag{17}\]  
Then, the partial derivative of \(L(\mathbf{P})\) with respect to \(\mathbf{P}\) equals:  
\[