# Singular Value Thresholding Algorithm for Wireless Sensor Network Localization

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Background

#### 2.1. Range-Based Localization

#### 2.2. Trilateration

## 3. The Singular Value Thresholding Algorithm

**Definition**

**1.**

**Theorem**

**1.**

## 4. Simulation

- Calculate the distance between coordinate p and anchor node, ${d}_{i}=\Vert p-{q}_{i}\Vert $. Then, we have$${\Vert p-{q}_{i}\Vert}^{2}={\Vert p\Vert}^{2}-2\langle {q}_{i},p\rangle +{\Vert {q}_{i}\Vert}^{2}={d}_{i}{}^{2}$$Subtracting the equations, we get a system of three linear equations with three unknowns (the entries of p);$$A=2\left(\begin{array}{c}{({q}_{2}-{q}_{1})}^{T}\\ {({q}_{3}-{q}_{1})}^{T}\\ {({q}_{4}-{q}_{1})}^{T}\end{array}\right),b=\left(\begin{array}{c}{d}_{1}{}^{2}-{d}_{2}{}^{2}+{\Vert {q}_{2}\Vert}^{2}-{\Vert {q}_{1}\Vert}^{2}\\ {d}_{1}{}^{2}-{d}_{3}{}^{2}+{\Vert {q}_{3}\Vert}^{2}-{\Vert {q}_{1}\Vert}^{2}\\ {d}_{1}{}^{2}-{d}_{4}{}^{2}+{\Vert {q}_{4}\Vert}^{2}-{\Vert {q}_{1}\Vert}^{2}\end{array}\right)$$
- Solve p by solving $A(X)=b$.
- The results of p are transformed into the Euclidean Distance Matrix (EDM).

#### 4.1. Matrix Completion

#### 4.2. Nuclear Norm Minimization (NNM)

#### 4.2.1. Semidefinite Programming (SDP)

#### 4.2.2. Singular Value Thresholding

## 5. Results and Discussions

- Next, matrix completion is implemented using Singular Value Thresholding in MATLAB and the results are attached in Appendix 1.
- The complete EDM is now reconstructed via the technique of Trilateration in MATLAB and the results are stated in Table 1.

^{−4}and 5.06 × 10

^{1}as the average relative error of EDM and the average relative recovery error, respectively. However, as stated in [35], where the best SDP solvers only work for matrices with sizes of less than 100 × 100, it can be seen that the relative error of EDM and relative recovery error of the reconstructed node location are high for sensor nodes with 100 or more nodes. Therefore, although the SDP method is simple and user-friendly with the application of CVX toolbox in MATLAB, the method is proven to have limitations in the number of sensor nodes, which affects the size of the EDM.

^{−1}, while the average relative error for matrix completion via SVT is 5.074 × 10

^{−5}.

## 6. Conclusions

^{−5}. However, the average relative recovery error of the reconstructed node location is 5.974 × 10

^{−1}which can be improvised, referring to the lower matrix completion’s relative error. Further work is needed to improvise the process of node reconstruction.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**Sensors and the partial Euclidean Distance Matrix (EDM).

**Left**: black dots represent the sensors, blue circles represent the objects, while red lines connect the neighboring sensors (nodes).

**Right:**blue dots represent the entries in the Euclidean Distance Matrix, corresponding to the neighboring sensors (nodes).

**Figure 3.**Graphs of the effects of object size to the proportion of missing entries for object size equal to 5, 10, 15, and 20 cm.

**Table 1.**Experimental results for 2-dimensional Wireless Sensor Network localization via Semidefinite Programming (CVX Toolbox).

Nu. of Sensor Node | Percentage of Missing Entries (%) | Relative Error on EDM | Relative Recovery Error | Processing Time (s) |
---|---|---|---|---|

10 | 20 | 4.89 × 10^{−5} | 4.31 × 10^{−1} | 0.04872 |

20 | 25 | 6.21 × 10^{−4} | 5.23 × 10^{−1} | 0.08231 |

50 | 40 | 6.89 × 10^{−4} | 5.65 × 10^{−1} | 1.14435 |

100 | 60 | 5.23 × 10^{4} | 3.57 × 10^{5} | 2.23154 |

200 | 80 | 4.23 × 10^{4} | 5.38 × 10^{7} | 6.5134 |

**Table 2.**Experimental results for 2-dimensional Wireless Sensor Network localization via Singular Value Thresholding.

Nu. of Sensor Node | Nu. of Iteration | Percentage of Missing Entries (%) | Relative Error on EDM | Relative Recovery Error | Processing Time (s) |
---|---|---|---|---|---|

10 | 17 | 26 | 5.64 × 10^{−5} | 6.26 × 10^{−1} | 0.036282 |

20 | 11 | 24 | 5.27 × 10^{−5} | 6.51 × 10^{−1} | 0.039764 |

50 | 9 | 22.4 | 3.82 × 10^{−5} | 6.24 × 10^{−1} | 0.081020 |

100 | 7 | 18.1 | 4.85 × 10^{−5} | 5.59 × 10^{−1} | 0.637559 |

200 | 7 | 19.1 | 5.77 × 10^{−5} | 5.27 × 10^{−1} | 2.720775 |

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**MDPI and ACS Style**

Ahmad Najib, Y.N.; Daud, H.; Abd Aziz, A.
Singular Value Thresholding Algorithm for Wireless Sensor Network Localization. *Mathematics* **2020**, *8*, 437.
https://doi.org/10.3390/math8030437

**AMA Style**

Ahmad Najib YN, Daud H, Abd Aziz A.
Singular Value Thresholding Algorithm for Wireless Sensor Network Localization. *Mathematics*. 2020; 8(3):437.
https://doi.org/10.3390/math8030437

**Chicago/Turabian Style**

Ahmad Najib, Yasmeen Nadhirah, Hanita Daud, and Azrina Abd Aziz.
2020. "Singular Value Thresholding Algorithm for Wireless Sensor Network Localization" *Mathematics* 8, no. 3: 437.
https://doi.org/10.3390/math8030437