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# A penalty decomposition method for nuclear norm minimization with l1 norm fidelity term

This work of Y. Shang is supported by the National Natural Science Foundation of China(No.11471102). This work of Z.-F Jin is supported by the National Natural Science Foundation of China(No.61370220) and the Plan for Scientific Innovation Talent of Henan Province(No.174200510011). This work of Duo Wang is supported by the National Natural Science Foundation of China(No.11701150)

• It is well known that the nuclear norm minimization problems are widely used in numerous fields such as machine learning, system recognition, and image processing, etc. It has captured considerable attention and taken good progress. Many researchers have made great contributions to the nuclear norm minimization problem with $l_{2}$ norm fidelity term, which is just able to deal with those problems with Gaussian noise. In this paper, we propose an efficient penalty decomposition(PD) method to solve the nuclear norm minimization problem with $l_{1}$ norm fidelity term. One subproblem admits a closed-form solution due to its special structure, and another can also get a closed-form solution by linearizing its quadratic term. The convergence results of the proposed method are given as well. Finally, the effectiveness and efficiency of the proposed method are verified by some numerical experiments.

Mathematics Subject Classification: Primary: 90C06, 90C25, 90C30.

 Citation: • • Figure 1.  Numerical comparisons of model (22) and (23) with three noisy cases(m = n = 500, r = 10, sr = 0.5)

Figure 2.  Numerical comparisons of model (22) and (23) with noiseless(m = n = 500). First row: $r = 10$, $sr$ is $0.3,0.5,0.7$ respectively. Second row: $sr = 0.5$, $r$ is $5,15,30$ respectively

Figure 3.  (a): Original gray image($512\times 512$); (b): Corresponding low rank images with $r = 30$; (c): Randomly masked image from rank $30$ with $sr = 70\%$ and $q = 0.01$; (e):Randomly masked image from rank $30$ with $sr = 70\%$ and $\sigma = 0.001$, $q = 0.01$; (d), (f): Recovered images by $PD-l_{1}$, whose "RelErr" are $4.760e-3$, $4.827e-3$ respectively

Figure 4.  Numerical results of $PD-l_{1}$ for problem (7) with impulsive noise: m = n = 500, r = 10, sr = 0.5, 0.7, 0.9, q = 0.01

Table 1.  Numerical results of $PD-l_{1}$ for (7), m = n, sr = 0.5, q = 0.01

 $PD-l_{1}$ $(m, r)$ p/dr Iter Time RelErr $(100, 5)$ 5.13 392 2.84 9.905e-004 $(200, 5)$ 10.13 187 4.03 9.881e-004 $(300, 5)$ 15.13 130 6.51 9.916e-004 $(400, 5)$ 20.13 99 9.26 9.947e-004 $(500, 5)$ 25.13 91 14.44 9.709e-004 $(600, 5)$ 30.13 78 19.31 9.853e-004 $(700, 5)$ 35.13 85 36.56 9.984e-004 $(800, 5)$ 40.13 88 54.63 9.889e-004 $(900, 5)$ 45.13 60 51.40 9.793e-004 $(1000, 5)$ 50.13 62 72.48 9.737e-004

Table 2.  Numerical results of $PD-l_{1}$ for (7), m = n, sr = 0.5

 $PD-l_{1}$ $(m, r)$ p/dr q $\sigma$ Time RelErr $(500, 10)$ 12.63 0.01 0.00 21.43 9.905e-004 $(500, 10)$ 12.63 0.01 0.001 21.65 9.822e-004 $(500, 10)$ 12.63 0.01 0.0001 21.77 9.781e-004 $(500, 10)$ 12.63 0.001 0.0001 20.61 8.054e-004 $(500, 10)$ 12.63 0.005 0.0001 20.53 8.659e-004 $(500, 10)$ 12.63 0.00 0.0001 2.15 7.722e-005
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