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Nonconvex mixed matrix minimization

  • * Corresponding author: Xianchao Xiu

    * Corresponding author: Xianchao Xiu 
This work was supported in part by the National Natural Science Foundation of China (61633001, 11671029, 11601348).
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  • Given the ultrahigh dimensionality and the complex structure, which contains matrices and vectors, the mixed matrix minimization becomes crucial for the analysis of those data. Recently, the nonconvex functions such as the smoothly clipped absolute deviation, the minimax concave penalty, the capped $ \ell_1 $-norm penalty and the $ \ell_p $ quasi-norm with $ 0<p<1 $ have been shown remarkable advantages in variable selection due to the fact that they can overcome the over-penalization. In this paper, we propose and study a novel nonconvex mixed matrix minimization, which combines the low-rank and sparse regularzations and nonconvex functions perfectly. The augmented Lagrangian method (ALM) is proposed to solve the noncovnex mixed matrix minimization problem. The resulting subproblems either have closed-form solutions or can be solved by fast solvers, which makes the ALM particularly efficient. In theory, we prove that the sequence generated by the ALM converges to a stationary point when the penalty parameter is above a computable threshold. Extensive numerical experiments illustrate that our proposed nonconvex mixed matrix minimization model outperforms the existing ones.

    Mathematics Subject Classification: Primary: 90C26, 90C46; Secondary: 90C90, 65K10.

    Citation:

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  • Table 1.  Comparison Results for Synthetic Data

    Cases RMSE(B) RMSE($ \gamma $)
    $ r $ $ s $ LMM LSMM NonMM LMM LSMM NonMM
    5 0.01 0.0000 0.0000 0.0000 0.1806 0.1421 0.1101
    0.05 0.0456 0.0437 0.0274 0.6453 0.2742 0.2336
    0.1 0.0905 0.0688 0.0496 0.9162 0.3788 0.3176
    0.2 0.1314 0.1082 0.0911 1.0822 0.5063 0.4410
    0.5 0.2744 0.2152 0.1013 1.5868 0.7371 0.7092
    10 0.01 0.0000 0.0000 0.0000 0.1927 0.1513 0.1217
    0.05 0.0781 0.05115 0.0382 0.7663 0.3312 0.2961
    0.1 0.1072 0.0749 0.0536 1.0452 0.3879 0.3401
    0.2 0.1339 0.1217 0.1161 1.2696 0.5305 0.4918
    0.5 0.2947 0.2440 0.1278 1.9561 0.7971 0.7330
    15 0.01 0.0000 0.0000 0.0000 0.2197 0.1920 0.1405
    0.05 0.0982 0.0749 0.0654 0.8408 0.3223 0.2893
    0.1 0.1323 0.1130 0.1108 1.2501 0.3534 0.3293
    0.2 0.1846 0.1560 0.1272 1.6730 0.4988 0.4494
    0.5 0.3176 0.2812 0.2176 2.3608 0.7817 0.7502
    20 0.01 0.0000 0.0000 0.0000 0.1936 0.1532 0.1509
    0.05 0.0987 0.0791 0.0732 0.9081 0.3168 0.2719
    0.1 0.1528 0.1454 0.1402 1.416 0.4583 0.3961
    0.2 0.1904 0.1821 0.1723 1.8304 0.4722 0.4148
    0.5 0.3876 0.2922 0.2134 2.4961 0.8049 0.7891
    30 0.01 0.0000 0.0000 0.0000 0.1768 0.1620 0.1400
    0.05 0.1071 0.0932 0.0858 1.0089 0.2360 0.2232
    0.1 0.1608 0.1572 0.1469 1.5286 0.3241 0.3062
    0.2 0.2487 0.2371 0.2206 2.082 0.4920 0.4599
    0.5 0.4170 0.3943 0.3725 2.6203 0.7856 0.8041
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    Table 2.  Comparison Results for Real-world Data

    Rate LMM LSMM NonMM
    5-fold 0.1321 0.1178 0.1025
    10-fold 0.1972 0.1630 0.1589
     | Show Table
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