Algorithm 2. Anomaly-aware Node Localization (ANLoC) Algorithm

Input: The sampled EDM matrix $\mathit{M}$, the index set $\mathsf{\Omega }$, the parameters ${\lambda }_1$, ${\lambda }_2$ and threshold $\theta$, the initial number of Gaussian components $n_c$. The coordinates of anchor nodes $\left\{{\mathit{a}}_1,{\mathit{a}}_2,\dots ,{\mathit{a}}_{n_a}\right\}\left(n_a\ge d+1\right),$ where $n_a$ denotes the number of anchor nodes.

Output: Coordinates of all unknown nodes $\left\{{\mathit{a}}_i|i=n_a+1,n_a+2,\dots n\right\}$.

1. Calculate the true underlying EDM matrix $\widehat{\mathit{D}}$ by using Algorithm 1;

2. Double centering the matrix $\widehat{\mathit{D}}$:

$\mathit{S}=-0.5\times \mathit{J}\widehat{\mathit{D}}\mathit{J}$, where $\mathit{J}=\mathit{I}-1·1^T/n$ and $\mathit{I}$ is identity matrix;

3. Perform SVD decomposition on matrix $\mathit{S}$:

$\left[\mathit{H},\mathit{\Lambda },\mathit{{\rm K}}\right]=svd\left(\mathit{S}\right)$;

4. Calculate relative coordinates:

$\mathit{T}=\left[{\mathit{t}}_1,{\mathit{t}}_2,\dots ,{\mathit{t}}_n\right]=\sqrt{{\mathit{\Lambda }}_d}·{\mathit{H}}_d^T$

where ${\mathit{t}}_i\in {ℝ}^{d\times 1}$, ${\mathit{\Lambda }}_d= \mathit{\Lambda }\left(1:d,1:d\right)$, ${\mathit{H}}_d=\mathit{H}\left(:,1:d\right)$;

5. Calculate the coordinate transformation matrix $\mathit{Q}$:

$\mathit{Q}=\left[{\mathit{a}}_2-{\mathit{a}}_1,{\mathit{a}}_3-{\mathit{a}}_1,\dots ,{\mathit{a}}_{n_a}-{\mathit{a}}_1\right]/\left[{\mathit{t}}_2-{\mathit{t}}_1,{\mathit{t}}_3-{\mathit{t}}_1,\dots ,{\mathit{t}}_{n_a}-{\mathit{t}}_1\right]$;

6. Calculate and output coordinates of all unknown nodes:

${\mathit{a}}_i=\mathit{Q}·\left({\mathit{t}}_i-{\mathit{t}}_1\right)+{\mathit{a}}_1,i=n_a+1,n_a+2,\dots n.$