# An Accelerated Symmetric Nonnegative Matrix Factorization Algorithm Using Extrapolation

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## Abstract

**:**

## 1. Introduction

## 2. Accelerated SNMF Algorithm

#### 2.1. Multiplicative Update Algorithm for SNMF

**Definition**

**1**

Algorithm 1:$\mathit{G}$ = MU-SNMF($\mathit{A}$, r). |

Step 1: InitializationInitialize ${\mathit{G}}^{0}$. Set $t=0$. Step 2: Update stagerepeat${\mathit{G}}^{t+1}={\mathit{G}}^{t}\otimes \sqrt[3]{{\displaystyle \frac{\mathit{A}{\mathit{G}}^{t}}{{\mathit{G}}^{t}{{\mathit{G}}^{t}}^{T}{\mathit{G}}^{t}}}}$ $t=t+1$ until the stopping criterion is satisfiedStep 3: Output $\mathit{G}\leftarrow {\mathit{G}}^{t}$.⊗ and ${\scriptstyle \frac{[\phantom{\rule{4pt}{0ex}}\xb7\phantom{\rule{4pt}{0ex}}]}{[\phantom{\rule{4pt}{0ex}}\xb7\phantom{\rule{4pt}{0ex}}]}}$ denote elementwise product and division, respectively. |

#### 2.2. Nesterov’s Accelerated Gradient

#### 2.3. Accelerated MU-SNMF Algorithm

Algorithm 2:$\mathit{G}$ = AMU-SNMF($\mathit{A}$, r). |

Step 1: InitializationInitialize ${\mathit{G}}^{0}$. Set $t=0$, ${t}^{restart}=0$, $\epsilon ={10}^{-16}$, ${F}^{0}={\Vert \mathit{A}-{\mathit{G}}^{0}{{\mathit{G}}^{0}}^{T}\Vert}_{F}^{2}$. Step 2: Update stagerepeat$\gamma}^{t}=1-\frac{3}{5+t-{t}^{restart}$ if$\phantom{\rule{4pt}{0ex}}t={t}^{restart}$ then${\mathit{Y}}^{t+1}={\mathit{G}}^{t}$ else${\mathit{Y}}^{t+1}=\mathrm{max}\{(1+{\gamma}^{t}){\mathit{G}}^{t}-{\gamma}^{t}{\mathit{G}}^{t-1},\epsilon \}$ end if${\mathit{G}}^{new}={\mathit{Y}}^{t+1}\otimes \sqrt[3]{{\displaystyle \frac{\mathit{A}{\mathit{Y}}^{t+1}}{{\mathit{Y}}^{t+1}{{\mathit{Y}}^{t+1}}^{T}{\mathit{Y}}^{t+1}}}}$ ${F}^{new}={\Vert \mathit{A}-{\mathit{G}}^{new}{\left({\mathit{G}}^{new}\right)}^{T}\Vert}_{F}^{2}$ if $\phantom{\rule{4pt}{0ex}}{F}^{new}>{F}^{t}$ then${t}^{restart}=t+1$ ${\mathit{G}}^{new}={\mathit{G}}^{t}$ ${F}^{new}={F}^{t}$ end if${\mathit{G}}^{t+1}={\mathit{G}}^{new}$ ${F}^{t+1}={F}^{new}$ $t=t+1$ until the stopping criterion is satisfiedStep 3: Output $\mathit{G}\leftarrow {\mathit{G}}^{t}$.⊗ and ${\scriptstyle \frac{[\phantom{\rule{4pt}{0ex}}\xb7\phantom{\rule{4pt}{0ex}}]}{[\phantom{\rule{4pt}{0ex}}\xb7\phantom{\rule{4pt}{0ex}}]}}$ denote elementwise product and division, respectively. |

#### 2.4. Symmetric Nonnegative Tensor Factorization (SNTF)

## 3. Experiments and Results

#### 3.1. Synthetic Data

#### 3.2. Document Clustering

#### 3.3. Object Clustering

#### 3.4. SNTF

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**The trajectories of $J\left(\mathit{G}\right)$ generated by the averaging approach on a three order symmetric nonnegative tensor $\mathcal{A}\in {\mathbb{R}}_{+}^{10\times 10\times 10}$, where 4 different SNMF algorithms were used separately in the averaging approach.

**Figure 2.**The plots of the average $E\left(\mathit{G}\right)$ versus runtime on the synthetic data. (

**a**) Noise-free. (

**b**) $\mathrm{SNR}\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}10$ dB.

**Figure 3.**The plots of the average $F\left(\mathit{G}\right)$ versus runtime on (

**a**) TDT2 and (

**b**) Reuters-21578.

TDT2 | Category | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 |

Number of documents | 167 | 160 | 145 | 141 | 140 | 131 | 123 | 123 | 120 | 104 | |

Reuters-21578 | Category | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 |

Number of documents | 63 | 60 | 53 | 45 | 45 | 44 | 42 | 38 | 38 | 37 |

**Table 2.**The mean of CA and the mean of NMI of the SNMF algorithms on the TDT2 and the Reuters-21578 collections.

MU-SNMF [6] | $\mathit{\alpha}$-SNMF [6] | $\mathit{\beta}$-SNMF [6] | Newton-Like SNMF [10] | vBSUM [8] | sBSUM [8] | BCD [7] | AMU-SNMF | ||
---|---|---|---|---|---|---|---|---|---|

TDT2 | CA | 0.9266 | 0.9035 | 0.9187 | 0.9348 | 0.9299 | 0.9312 | 0.9133 | 0.9631 |

NMI | 0.9363 | 0.9197 | 0.9292 | 0.9477 | 0.9450 | 0.9464 | 0.9354 | 0.9625 | |

Reuters-21578 | CA | 0.6711 | 0.6803 | 0.6786 | 0.6778 | 0.6671 | 0.6630 | 0.6749 | 0.6877 |

NMI | 0.6329 | 0.6380 | 0.6383 | 0.6408 | 0.6365 | 0.6345 | 0.6389 | 0.6449 |

MU-SNMF [6] | $\mathit{\alpha}$-SNMF [6] | $\mathit{\beta}$-SNMF [6] | Newton-Like SNMF [10] | vBSUM [8] | sBSUM [8] | BCD [7] | AMU-SNMF | ||
---|---|---|---|---|---|---|---|---|---|

COIL-20 | CA | 0.6641 | 0.6527 | 0.6890 | 0.7247 | 0.7086 | 0.6657 | 0.6362 | 0.7332 |

NMI | 0.7826 | 0.7774 | 0.8160 | 0.8492 | 0.8454 | 0.8166 | 0.7882 | 0.8554 |

**Table 4.**The mean of $Fit\left(\mathit{G}\right)$ of the averaging approach using different SNMF algorithms.

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**MDPI and ACS Style**

Wang, P.; He, Z.; Lu, J.; Tan, B.; Bai, Y.; Tan, J.; Liu, T.; Lin, Z.
An Accelerated Symmetric Nonnegative Matrix Factorization Algorithm Using Extrapolation. *Symmetry* **2020**, *12*, 1187.
https://doi.org/10.3390/sym12071187

**AMA Style**

Wang P, He Z, Lu J, Tan B, Bai Y, Tan J, Liu T, Lin Z.
An Accelerated Symmetric Nonnegative Matrix Factorization Algorithm Using Extrapolation. *Symmetry*. 2020; 12(7):1187.
https://doi.org/10.3390/sym12071187

**Chicago/Turabian Style**

Wang, Peitao, Zhaoshui He, Jun Lu, Beihai Tan, YuLei Bai, Ji Tan, Taiheng Liu, and Zhijie Lin.
2020. "An Accelerated Symmetric Nonnegative Matrix Factorization Algorithm Using Extrapolation" *Symmetry* 12, no. 7: 1187.
https://doi.org/10.3390/sym12071187