# A Unified Proximity Algorithm with Adaptive Penalty for Nuclear Norm Minimization

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

## 3. Proximity Algorithm with Adaptive Penalty

#### 3.1. Proximity Algorithm

**Lemma**

**1.**

**Proof.**

#### 3.2. Adaptive Penalty

Algorithm 1 PA-AP for solving (5) |

Input: Observation vector $\mathbf{b}$, linear mapping $\mathcal{A}$, and some parameters ${\mu}_{0},{\rho}_{0},{\epsilon}_{1},{\epsilon}_{2}>0$, ${\mu}_{max}\gg {\mu}_{0}>0$. |

Initialize: Set ${X}_{0}$ and ${\mathbf{y}}_{0}$ to zero matrix and vector, respectively. Set $\mu ={\mu}_{0}$ and ${\eta >\parallel \mathcal{A}\parallel}_{2}^{2}$. Set $k=0$. |

while not converged, do |

step 1: Update ${X}_{k+1}$, ${\mathbf{y}}_{k+1}$ and ${\lambda}_{k+1}$ in turn by (30). |

step 2: Update ${\mu}_{k+1}$ by (34), and let $\mu \leftarrow {\mu}_{k+1}$. |

step 3: $k\leftarrow k+1$. |

end while |

#### 3.3. Convergence

- (i)
- ${M}_{0}={\sum}_{i=1}^{l}{M}_{i}$,
- (ii)
- ${M}_{1}={M}_{2}=\cdots ={M}_{l-1}$,
- (iii)
- $H:={M}_{0}+{M}_{l}$, H is symmetric positive definite,
- (iv)
- $\mathcal{N}\left(H\right)\subseteq \mathcal{N}\left({M}_{l}\right)\cap \mathcal{N}\left({M}_{l}^{T}\right)$,
- (v)
- $\parallel {\left({H}^{\u2020}\right)}^{\frac{1}{2}}{M}_{l}{\left({H}^{\u2020}\right)}^{\frac{1}{2}}{\parallel}_{2}<\frac{1}{2}$,

**Theorem**

**1.**

**Proof.**

**Corollary**

**1.**

**Proof.**

## 4. Numerical Experiments

#### 4.1. Nuclear Norm Minimization Problem

#### 4.2. Matrix Completion

#### 4.3. Low-Rank Image Recovery

## 5. Conclusions and Future Work

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Original $512\times \phantom{\rule{3.33333pt}{0ex}}512$ images (Lena, Pirate, Cameraman) with full rank (first column); Corresponding low rank images with $r=40$ (second column); Randomly masked images from rank 40 images with $sr=40\%$ (third column); Recovered images by PA-AP (last column).

**Figure 2.**Convergence behavior of the four methods ($Lena,sr=0.4,{\epsilon}_{2}={10}^{-5}$). The first subfigure is the estimated rank; the second is the relative error to the original matrix; and the last is the running time.

(n, r) | p/dof | sr | Prob. (3) ($\mathit{\delta}$ = 0) | Prob. (3) ($\mathit{\delta}$ = 10^{−2}) | Prob. (4) | ||||||
---|---|---|---|---|---|---|---|---|---|---|---|

Iter | Time | RelErr | Iter | Time | RelErr | Iter | Time | RelErr | |||

(128, 3) | 4.32 | 0.2 | 86 | 2.03 | 3.942 × 10^{−3} | 83 | 1.83 | 5.706 × 10^{−3} | 83 | 2.14 | 8.050 × 10^{−3} |

(128, 3) | 8.64 | 0.4 | 31 | 0.57 | 4.042 × 10^{−4} | 33 | 0.69 | 5.535 × 10^{−3} | 34 | 0.70 | 6.196 × 10^{−3} |

(128, 3) | 12.95 | 0.6 | 20 | 0.44 | 1.376 × 10^{−4} | 20 | 0.45 | 5.652 × 10^{−3} | 20 | 0.48 | 5.714 × 10^{−3} |

(128, 3) | 17.27 | 0.8 | 13 | 0.29 | 6.251 × 10^{−5} | 12 | 0.34 | 5.952 × 10^{−3} | 13 | 0.28 | 5.966 × 10^{−3} |

(256, 5) | 5.17 | 0.2 | 53 | 2.04 | 3.178 × 10^{−4} | 53 | 2.02 | 3.846 × 10^{−3} | 53 | 2.08 | 4.353 × 10^{−3} |

(256, 5) | 10.34 | 0.4 | 31 | 1.22 | 1.649 × 10^{−4} | 31 | 1.16 | 4.424 × 10^{−3} | 31 | 1.28 | 4.404 × 10^{−3} |

(256, 5) | 15.51 | 0.6 | 20 | 0.78 | 1.139 × 10^{−4} | 20 | 0.78 | 4.414 × 10^{−3} | 20 | 0.76 | 4.419 × 10^{−3} |

(256, 5) | 20.68 | 0.8 | 13 | 0.50 | 3.893 × 10^{−5} | 13 | 0.52 | 4.458 × 10^{−3} | 14 | 0.52 | 4.297 × 10^{−3} |

(512, 10) | 5.17 | 0.2 | 55 | 7.63 | 2.397 × 10^{−4} | 55 | 7.77 | 2.858 × 10^{−3} | 54 | 7.86 | 2.939 × 10^{−3} |

(512, 10) | 10.34 | 0.4 | 34 | 4.22 | 1.259 × 10^{−4} | 34 | 4.55 | 3.068 × 10^{−3} | 34 | 4.46 | 3.081 × 10^{−3} |

(512, 10) | 15.51 | 0.6 | 22 | 2.85 | 1.195 × 10^{−4} | 22 | 2.98 | 3.106 × 10^{−3} | 22 | 2.96 | 3.177 × 10^{−3} |

(512, 10) | 20.68 | 0.8 | 13 | 1.81 | 1.009 × 10^{−4} | 13 | 1.85 | 3.129 × 10^{−3} | 13 | 1.86 | 3.262 × 10^{−3} |

(n, r) | p/dof | sr | PA-AP | IADM-CG | IADM-BB | ||||||
---|---|---|---|---|---|---|---|---|---|---|---|

Iter | Time | RelErr | Iter | Time | RelErr | Iter | Time | RelErr | |||

(500, 10) | 5.05 | 0.2 | 54 | 8.65 | 2.881 × 10^{−3} | 33 | 15.09 | 8.312 × 10^{−3} | 39 | 26.65 | 9.448 × 10^{−3} |

(500, 10) | 10.10 | 0.4 | 33 | 4.14 | 3.035 × 10^{−3} | 23 | 8.45 | 4.433 × 10^{−3} | 26 | 16.58 | 6.484 × 10^{−3} |

(500, 10) | 15.15 | 0.6 | 22 | 2.15 | 3.20 × 10^{−3} | 15 | 5.56 | 3.968 × 10^{−3} | 18 | 9.81 | 4.581 × 10^{−3} |

(500, 10) | 20.20 | 0.8 | 13 | 1.72 | 3.224 × 10^{−3} | 10 | 4.79 | 3.292 × 10^{−3} | 12 | 7.50 | 3.528 × 10^{−3} |

(1000, 20) | 5.05 | 0.2 | 70 | 38.20 | 2.110 × 10^{−3} | 33 | 69.23 | 8.806 × 10^{−3} | 40 | 141.57 | 5.047 × 10^{−3} |

(1000, 20) | 10.10 | 0.4 | 38 | 15.14 | 2.379 × 10^{−3} | 22 | 41.34 | 3.442 × 10^{−3} | 23 | 78.78 | 7.097 × 10^{−3} |

(1000, 20) | 15.15 | 0.6 | 21 | 8.83 | 2.322 × 10^{−3} | 17 | 30.52 | 3.362 × 10^{−3} | 19 | 60.35 | 4.277 × 10^{−3} |

(1000, 20) | 20.20 | 0.8 | 14 | 6.60 | 2.379 × 10^{−3} | 13 | 25.36 | 3.665 × 10^{−3} | 20 | 57.51 | 2.724 × 10^{−3} |

(2000, 20) | 5.05 | 0.2 | 74 | 144.89 | 2.270 × 10^{−3} | 39 | 581.64 | 1.039 × 10^{−2} | 46 | 1399.31 | 8.316 × 10^{−3} |

(2000, 20) | 10.10 | 0.4 | 33 | 62.68 | 2.348 × 10^{−3} | 22 | 328.02 | 4.755 × 10^{−3} | 22 | 628.66 | 5.090 × 10^{−3} |

(2000, 20) | 15.15 | 0.6 | 21 | 34.60 | 2.368 × 10^{−3} | 14 | 172.70 | 3.405 × 10^{−3} | 17 | 438.01 | 4.527 × 10^{−3} |

(2000, 20) | 20.20 | 0.8 | 13 | 23.40 | 2.401 × 10^{−3} | 20 | 128.73 | 2.752 × 10^{−3} | 15 | 359.59 | 3.260 × 10^{−3} |

(n, r) | p/dof | sr | Prob. (3) ($\mathit{\delta}$ = 0) | Prob. (3) ($\mathit{\delta}$ = 10^{−2}) | Prob. (4) | ||||||
---|---|---|---|---|---|---|---|---|---|---|---|

Iter | Time | RelErr | Iter | Time | RelErr | Iter | Time | RelErr | |||

(1000, 10) | 10.05 | 0.2 | 76 | 21.94 | 2.439 × 10^{−5} | 76 | 21.52 | 3.015 × 10^{−3} | 76 | 22.92 | 5.245 × 10^{−4} |

(1000, 10) | 20.10 | 0.4 | 40 | 9.28 | 1.159 × 10^{−5} | 40 | 9.68 | 3.085 × 10^{−3} | 39 | 9.55 | 7.436 × 10^{−4} |

(1000, 10) | 30.15 | 0.6 | 22 | 5.72 | 1.061 × 10^{−6} | 23 | 6.36 | 3.105 × 10^{−3} | 23 | 6.10 | 8.251 × 10^{−4} |

(1000, 10) | 40.20 | 0.8 | 12 | 3.88 | 2.859 × 10^{−6} | 14 | 3.76 | 3.075 × 10^{−3} | 14 | 3.74 | 8.668 × 10^{−4} |

(1000, 20) | 5.05 | 0.2 | 87 | 39.10 | 3.037 × 10^{−5} | 88 | 43.58 | 1.995 × 10^{−3} | 87 | 43.71 | 5.469 × 10^{−4} |

(1000, 20) | 10.10 | 0.4 | 46 | 13.02 | 1.277 × 10^{−5} | 46 | 13.69 | 2.115 × 10^{−3} | 46 | 13.47 | 7.467 × 10^{−4} |

(1000, 20) | 15.15 | 0.6 | 25 | 7.56 | 1.230 × 10^{−5} | 25 | 7.54 | 2.158 × 10^{−3} | 25 | 7.64 | 8.232 × 10^{−4} |

(1000, 20) | 20.20 | 0.8 | 14 | 4.86 | 9.617 × 10^{−6} | 14 | 4.87 | 2.194 × 10^{−3} | 14 | 4.88 | 8.624 × 10^{−4} |

(2000, 10) | 20.05 | 0.2 | 80 | 46.73 | 2.538 × 10^{−5} | 80 | 67.92 | 3.086 × 10^{−3} | 80 | 54.29 | 7.422 × 10^{−4} |

(2000, 10) | 40.10 | 0.4 | 37 | 26.02 | 1.299 × 10^{−5} | 37 | 31.65 | 3.091 × 10^{−3} | 37 | 27.20 | 8.641 × 10^{−4} |

(2000, 10) | 60.15 | 0.6 | 23 | 19.30 | 9.672 × 10^{−6} | 23 | 21.64 | 3.160 × 10^{−3} | 23 | 20.02 | 9.024 × 10^{−4} |

(2000, 10) | 80.20 | 0.8 | 13 | 11.99 | 2.584 × 10^{−6} | 13 | 13.31 | 3.153 × 10^{−3} | 13 | 11.71 | 9.276 × 10^{−4} |

(2000, 20) | 10.05 | 0.2 | 91 | 108.16 | 2.853 × 10^{−5} | 91 | 132.04 | 2.133 × 10^{−3} | 91 | 129.30 | 7.461 × 10^{−4} |

(2000, 20) | 20.10 | 0.4 | 40 | 42.54 | 1.146 × 10^{−5} | 40 | 44.62 | 2.190 × 10^{−3} | 40 | 42.49 | 8.674 × 10^{−4} |

(2000, 20) | 30.15 | 0.6 | 24 | 27.07 | 8.683 × 10^{−6} | 24 | 28.24 | 2.226 × 10^{−3} | 24 | 27.43 | 9.058 × 10^{−4} |

(2000, 20) | 40.20 | 0.8 | 13 | 15.81 | 3.772 × 10^{−6} | 13 | 17.19 | 2.205 × 10^{−3} | 13 | 16.18 | 9.268 × 10^{−4} |

(n, r) | p | p/dof | sr | PA-AP | ADM | IADM-CG | IADM-BB | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Iter | Time | RelErr | Iter | Time | RelErr | Iter | Time | RelErr | Iter | Time | RelErr | ||||

(1024, 5) | 209235 | 10.29 | 0.2 | 80 | 17.52 | 2.218 × 10^{−5} | 81 | 18.74 | 6.133 × 10^{−4} | 52 | 58.41 | 3.266 × 10^{−3} | 50 | 112.50 | 1.801 × 10^{−3} |

(1024, 5) | 314529 | 30.80 | 0.3 | 50 | 10.25 | 1.743 × 10^{−5} | 56 | 11.60 | 3.988 × 10^{−4} | 37 | 39.52 | 3.639 × 10^{−3} | 51 | 96.33 | 3.023 × 10^{−3} |

(1024, 5) | 419547 | 41.06 | 0.4 | 37 | 7.57 | 1.036 × 10^{−5} | 40 | 8.31 | 3.023 × 10^{−4} | 36 | 36.77 | 2.439 × 10^{−3} | 37 | 67.10 | 2.655 × 10^{−3} |

(1024, 5) | 525213 | 51.33 | 0.5 | 28 | 6.44 | 8.484 × 10^{−6} | 31 | 7.08 | 2.600 × 10^{−4} | 31 | 33.22 | 1.102 × 10^{−3} | 29 | 57.78 | 1.414 × 10^{−3} |

(1024, 5) | 628736 | 61.59 | 0.6 | 22 | 4.89 | 6.311 × 10^{−6} | 26 | 6.01 | 2.063 × 10^{−4} | 21 | 22.91 | 2.330 × 10^{−3} | 32 | 55.61 | 1.020 × 10^{−3} |

(1024, 5) | 733429 | 71.86 | 0.7 | 17 | 4.20 | 4.239 × 10^{−6} | 21 | 5.42 | 1.738 × 10^{−4} | 40 | 40.23 | 1.361 × 10^{−4} | 53 | 49.85 | 1.743 × 10^{−4} |

(1024, 5) | 838513 | 82.12 | 0.8 | 13 | 3.17 | 3.840 × 10^{−6} | 16 | 3.95 | 1.593 × 10^{−4} | 41 | 29.95 | 4.171 × 10^{−4} | 30 | 41.92 | 1.031 × 10^{−4} |

(1024, 5) | 943801 | 92.39 | 0.9 | 13 | 2.58 | 2.782 × 10^{−6} | 12 | 2.58 | 1.386 × 10^{−4} | 11 | 12.16 | 8.341 × 10^{−4} | 12 | 19.25 | 6.523 × 10^{−4} |

(1024, 10) | 209469 | 10.29 | 0.2 | 77 | 15.94 | 2.341 × 10^{−5} | 71 | 14.79 | 6.273 × 10^{−4} | 51 | 50.90 | 2.646 × 10^{−3} | 53 | 109.16 | 2.210 × 10^{−3} |

(1024, 10) | 315457 | 15.44 | 0.3 | 55 | 11.95 | 1.848 × 10^{−5} | 52 | 11.63 | 4.264 × 10^{−4} | 40 | 42.72 | 2.328 × 10^{−3} | 39 | 79.82 | 2.258 × 10^{−3} |

(1024, 10) | 419442 | 20.58 | 0.4 | 40 | 10.66 | 1.005 × 10^{−5} | 39 | 10.79 | 3.031 × 10^{−4} | 32 | 33.68 | 1.268 × 10^{−3} | 34 | 68.76 | 8.858 × 10^{−3} |

(1024, 10) | 524145 | 25.73 | 0.5 | 30 | 8.97 | 8.688 × 10^{−5} | 31 | 9.49 | 2.525 × 10^{−4} | 32 | 35.26 | 1.137 × 10^{−3} | 29 | 55.39 | 1.536 × 10^{−3} |

(1024, 10) | 629555 | 30.87 | 0.6 | 23 | 5.26 | 7.085 × 10^{−6} | 24 | 5.56 | 1.972 × 10^{−4} | 39 | 37.55 | 4.376 × 10^{−4} | 32 | 51.02 | 7.468 × 10^{−3} |

(1024, 10) | 733285 | 36.02 | 0.7 | 18 | 4.98 | 6.149 × 10^{−6} | 19 | 5.13 | 1.817 × 10^{−4} | 27 | 28.35 | 6.503 × 10^{−4} | 30 | 48.98 | 3.295 × 10^{−4} |

(1024, 10) | 838650 | 41.16 | 0.8 | 14 | 4.53 | 2.647 × 10^{−6} | 16 | 5.28 | 1.557 × 10^{−4} | 31 | 33.23 | 5.794 × 10^{−4} | 17 | 30.96 | 6.046 × 10^{−4} |

(1024, 10) | 943738 | 46.31 | 0.9 | 12 | 3.36 | 2.130 × 10^{−6} | 12 | 3.62 | 1.370 × 10^{−4} | 34 | 34.15 | 8.934 × 10^{−4} | 23 | 40.62 | 3.884 × 10^{−4} |

**Table 5.**Comparisons of PA-AP, ADM, IADM-CG and IADM-BB for noisy matrix completion ($\delta ={10}^{-2})$.

(n, r) | p | p/dof | sr | PA-AP | ADM | IADM-CG | IADM-BB | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Iter | Time | RelErr | Iter | Time | RelErr | Iter | Time | RelErr | Iter | Time | RelErr | ||||

(1024, 5) | 210151 | 20.53 | 0.2 | 61 | 16.96 | 4.345 × 10^{−3} | 60 | 13.61 | 4.354 × 10^{−3} | 40 | 95.66 | 9.546 × 10^{−3} | 45 | 237.45 | 5.234 × 10^{−3} |

(1024, 5) | 314332 | 30.80 | 0.3 | 43 | 10.68 | 4.413 × 10^{−3} | 43 | 10.88 | 4.448 × 10^{−3} | 27 | 64.23 | 1.215 × 10^{−2} | 31 | 164.52 | 5.319 × 10^{−3} |

(1024, 5) | 418708 | 41.06 | 0.4 | 31 | 8.95 | 4.416 × 10^{−3} | 33 | 9.50 | 4.419 × 10^{−3} | 25 | 59.75 | 6.410 × 10^{−3} | 24 | 123.38 | 5.318 × 10^{−3} |

(1024, 5) | 524429 | 51.33 | 0.5 | 24 | 7.65 | 4.387 × 10^{−3} | 25 | 7.74 | 4.402 × 10^{−3} | 21 | 49.00 | 4.920 × 10^{−3} | 19 | 97.81 | 4.791 × 10^{−3} |

(1024, 5) | 628736 | 61.59 | 0.6 | 19 | 5.25 | 4.452 × 10^{−3} | 19 | 4.93 | 4.452 × 10^{−3} | 19 | 40.69 | 5.035 × 10^{−3} | 14 | 73.53 | 5.177 × 10^{−3} |

(1024, 5) | 734131 | 71.86 | 0.7 | 15 | 4.47 | 4.335 × 10^{−3} | 17 | 4.56 | 4.337 × 10^{−3} | 17 | 39.24 | 4.791 × 10^{−3} | 21 | 76.87 | 4.524 × 10^{−3} |

(1024, 5) | 838476 | 82.12 | 0.8 | 12 | 3.64 | 4.444 × 10^{−3} | 14 | 3.97 | 4.446 × 10^{−3} | 19 | 39.52 | 1.184 × 10^{−3} | 15 | 54.21 | 4.479 × 10^{−3} |

(1024, 5) | 944170 | 92.39 | 0.9 | 10 | 3.16 | 4.486 × 10^{−3} | 11 | 2.94 | 4.557 × 10^{−3}> | 10 | 21.29 | 4.591 × 10^{−3} | 10 | 36.21 | 4.531 × 10^{−3} |

(1024, 10) | 210118 | 10.29 | 0.2 | 77 | 20.56 | 3.017 × 10^{−3} | 71 | 19.10 | 3.081 × 10^{−3} | 53 | 62.98 | 3.894 × 10^{−3} | 48 | 123.01 | 4.089 × 10^{−3} |

(1024, 10) | 314614 | 15.44 | 0.3 | 54 | 15.21 | 3.089 × 10^{−3} | 52 | 15.27 | 3.119 × 10^{−3} | 40 | 44.45 | 3.842 × 10^{−3} | 41 | 85.27 | 3.816 × 10^{−3} |

(1024, 10) | 420191 | 20.58 | 0.4 | 40 | 11.53 | 3.060 × 10^{−3} | 39 | 11.01 | 3.075 × 10^{−3} | 32 | 36.46 | 3.306 × 10^{−3} | 34 | 72.74 | 3.178 × 10^{−3} |

(1024, 10) | 523405 | 25.73 | 0.5 | 30 | 7.59 | 3.087 × 10^{−3} | 31 | 7.70 | 3.097 × 10^{−3} | 52 | 52.59 | 3.087 × 10^{−3} | 28 | 54.28 | 3.647 × 10^{−3} |

(1024, 10) | 628935 | 30.87 | 0.6 | 23 | 6.04 | 3.061 × 10^{−3} | 24 | 6.76 | 3.068 × 10^{−3} | 44 | 45.74 | 3.061 × 10^{−3} | 39 | 61.59 | 3.113 × 10^{−3} |

(1024, 10) | 734096 | 36.02 | 0.7 | 18 | 5.17 | 3.090 × 10^{−3} | 19 | 4.97 | 3.095 × 10^{−3} | 32 | 33.06 | 3.134 × 10^{−3} | 47 | 72.07 | 3.116 × 10^{−3} |

(1024, 10) | 838509 | 41.16 | 0.8 | 14 | 4.26 | 3.121 × 10^{−3} | 16 | 4.65 | 3.125 × 10^{−3} | 31 | 31.85 | 3.175 × 10^{−3} | 38 | 60.78 | 3.183 × 10^{−3} |

(1024, 10) | 944068 | 46.31 | 0.9 | 12 | 3.76 | 3.093 × 10^{−3} | 12 | 3.54 | 3.096 × 10^{−3} | 34 | 34.64 | 3.216 × 10^{−3} | 33 | 45.15 | 3.095 × 10^{−3} |

Name | sr | PA-AP | ADM | IADM-CG | IADM-BB | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Time | RelErr | PSNR | Time | RelErr | PSNR | Time | RelErr | PSNR | Time | RelErr | PSNR | ||

Lena | 0.2 | 58.66 | 1.730 × 10^{−4} | 26.17 | 127.86 | 1.751 × 10^{−4} | 26.61 | 148.71 | 1.625 × 10^{−4} | 26.78 | 161.59 | 1.633 × 10^{−4} | 26.78 |

0.4 | 46.09 | 3.984 × 10^{−5} | 84.67 | 94.65 | 4.578 × 10^{−5} | 85.52 | 94.86 | 1.191 × 10^{−4} | 76.06 | 110.74 | 1.244 × 10^{−4} | 75.58 | |

0.6 | 7.52 | 2.071 × 10^{−5} | 93.42 | 55.92 | 2.717 × 10^{−5} | 92.60 | 54.78 | 8.376 × 10^{−5} | 81.65 | 65.75 | 9.504 × 10^{−5} | 80.40 | |

0.8 | 3.41 | 1.025 × 10^{−5} | 101.59 | 37.32 | 1.771 × 10^{−5} | 99.04 | 33.46 | 4.374 × 10^{−5} | 89.62 | 39.09 | 6.070 × 10^{−5} | 86.65 | |

Pirate | 0.2 | 54.23 | 1.561 × 10^{−4} | 26.03 | 121.73 | 1.745 × 10^{−4} | 26.25 | 140.40 | 1.643 × 10^{−4} | 26.41 | 154.24 | 1.651 × 10^{−4} | 26.41 |

0.4 | 38.58 | 3.961 × 10^{−5} | 85.42 | 92.33 | 4.905 × 10^{−5} | 86.24 | 94.57 | 1.140 × 10^{−4} | 76.98 | 112.77 | 8.735 × 10^{−5} | 79.81 | |

0.6 | 8.15 | 1.313 × 10^{−5} | 97.97 | 53.13 | 2.839 × 10^{−5} | 92.92 | 54.81 | 4.376 × 10^{−5} | 86.89 | 66.42 | 6.227 × 10^{−5} | 84.26 | |

0.8 | 3.45 | 8.451 × 10^{−6} | 104.65 | 33.83 | 1.967 × 10^{−5} | 99.05 | 29.41 | 6.704 × 10^{−5} | 91.70 | 36.86 | 1.001 × 10^{−5} | 88.64 | |

Cameraman | 0.2 | 50.51 | 1.717 × 10^{−4} | 24.10 | 124.54 | 1.965 × 10^{−4} | 24.32 | 145.18 | 2.159 × 10^{−4} | 24.43 | 159.92 | 2.167 × 10^{−4} | 24.43 |

0.4 | 46.81 | 5.290 × 10^{−5} | 80.18 | 102.31 | 5.570 × 10^{−5} | 81.30 | 106.05 | 1.182 × 10^{−5} | 73.72 | 115.48 | 1.313 × 10^{−4} | 73.00 | |

0.6 | 10.03 | 2.389 × 10^{−5} | 90.92 | 59.87 | 3.090 × 10^{−5} | 90.36 | 58.43 | 6.099 × 10^{−5} | 83.07 | 71.95 | 5.343 × 10^{−5} | 84.40 | |

0.8 | 4.18 | 1.613 × 10^{−5} | 96.46 | 37.86 | 2.036 × 10^{−5} | 96.03 | 35.31 | 3.360 × 10^{−5} | 90.22 | 45.53 | 2.106 × 10^{−5} | 93.04 |

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## Share and Cite

**MDPI and ACS Style**

Hu, W.; Zheng, W.; Yu, G.
A Unified Proximity Algorithm with Adaptive Penalty for Nuclear Norm Minimization. *Symmetry* **2019**, *11*, 1277.
https://doi.org/10.3390/sym11101277

**AMA Style**

Hu W, Zheng W, Yu G.
A Unified Proximity Algorithm with Adaptive Penalty for Nuclear Norm Minimization. *Symmetry*. 2019; 11(10):1277.
https://doi.org/10.3390/sym11101277

**Chicago/Turabian Style**

Hu, Wenyu, Weidong Zheng, and Gaohang Yu.
2019. "A Unified Proximity Algorithm with Adaptive Penalty for Nuclear Norm Minimization" *Symmetry* 11, no. 10: 1277.
https://doi.org/10.3390/sym11101277