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In the era of big data, much of the data is susceptible to noise with heavy-tailed distribution. Fused Lasso can effectively handle high dimensional sparse data with strong correlation between two adjacent variables under known Gaussian noise. However, it has poor robustness to non-Gaussian noise with heavy-tailed distribution. Robust fused Lasso with l1 norm loss function can overcome the drawback of fused Lasso when noise is heavy-tailed distribution. But the key challenge for solving this model is nonsmoothness and its nonseparability. Therefore, in this paper, we first deform the robust fused Lasso into an easily solvable form, which changes the three-block objective function to a two-block form. Then, we propose an accelerated proximal alternating direction method of multipliers (APADMM) with an additional update step, which is base on a new PADMM that changes the Lagrangian multiplier term update. Furthermore, we give the O(1/K) nonergodic convergence rate analysis of the proposed APADMM. Finally, numerical results show that the proposed new PADMM and APADMM have better performance than other existing ADMM solvers. [ABSTRACT FROM AUTHOR]

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 l2 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 l1 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. [ABSTRACT FROM AUTHOR]

SUMMARY The aim of the nuclear norm minimization problem is to find a matrix that minimizes the sum of its singular values and satisfies some constraints simultaneously. Such a problem has received more attention largely because it is closely related to the affine rank minimization problem, which appears in many control applications including controller design, realization theory, and model reduction. In this paper, we first propose an exact version alternating direction method for solving the nuclear norm minimization problem with linear equality constraints. At each iteration, the method involves a singular value thresholding and linear matrix equations which are solved exactly. Convergence of the proposed algorithm is followed directly. To broaden the capacity of solving larger problems, we solve approximately the subproblem by an iterative method with the Barzilai-Borwein steplength. Some extensions to the noisy problems and nuclear norm regularized least-square problems are also discussed. Numerical experiments and comparisons with the state-of-the-art method FPCA show that the proposed method is effective and promising. Copyright © 2011 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]