Search

In recent years, most of the studies have shown that the generalized iterated shrinkage thresholdings (GISTs) have become the commonly used first-order optimization algorithms in sparse learning problems. The nonconvex relaxations of the $\ell _{0}$ -norm usually achieve better performance than the convex case (e.g., $\ell _{1}$ -norm) since the former can achieve a nearly unbiased solver. To increase the calculation efficiency, this work further provides an accelerated GIST version, that is, AGIST, through the extrapolation-based acceleration technique, which can contribute to reduce the number of iterations when solving a family of nonconvex sparse learning problems. Besides, we present the algorithmic analysis, including both local and global convergence guarantees, as well as other intermediate results for the GIST and AGIST, denoted as (A)GIST, by virtue of the Kurdyka-Łojasiewica (KŁ) property and some milder assumptions. Numerical experiments on both synthetic data and real-world databases can demonstrate that the convergence results of objective function accord to the theoretical properties and nonconvex sparse learning methods can achieve superior performance over some convex ones. [ABSTRACT FROM AUTHOR]

Recently, nuclear norm-based low rank representation (LRR) methods have been popular in several applications, such as subspace segmentation. However, there exist two limitations: one is that nuclear norm as the relaxation of rank function will lead to the suboptimal solution since nuclear norm-based minimization subproblem tends to the over-relaxations of singular value elements and treats each of them equally; the other is that solving LRR problems may cause more time consumption due to involving singular value decomposition of the large scale matrix at each iteration. To overcome both disadvantages, this paper mainly considers two tractable variants of LRR: one is Schatten- ${p}$ norm minimization-based LRR (i.e., ${S}_{p}$ NM_LRR) and the other is Schatten- ${p}$ norm factorization-based LRR (i.e., ${S}_{p}$ NF_LRR) for ${p=1}$ , 2/3 and 1/2. By introducing two or more auxiliary variables in the constraints, the alternating direction method of multiplier (ADMM) with multiple updating variables can be devised to solve these variants of LRR. Furthermore, both computational complexity and convergence property are given to evaluate nonconvex multiblocks ADMM algorithms. Several experiments finally validate the efficacy and efficiency of our methods on both synthetic data and real world data. [ABSTRACT FROM AUTHOR]

Face recognition (FR) via regression analysis-based classification has been widely studied in the past several years. Most existing regression analysis methods characterize the pixelwise representation error via l1 -norm or l2 -norm, which overlook the 2D structure of the error image. Recently, the nuclear norm-based matrix regression model is proposed to characterize low-rank structure of the error image. However, the nuclear norm cannot accurately describe the low-rank structural noise when the incoherence assumptions on the singular values does not hold, since it overpenalizes several much larger singular values. To address this problem, this paper presents the robust nuclear norm to characterize the structural error image and then extends it to deal with the mixed noise. The majorization–minimization (MM) method is applied to derive a iterative scheme for minimization of the robust nuclear norm optimization problem. Then, an efficiently alternating direction method of multipliers (ADMM) method is used to solve the proposed models. We use weighted nuclear norm as classification criterion to obtain the final recognition results. Experiments on several public face databases demonstrate the effectiveness of our models in handling with variations of structural noise (occlusion, illumination, and so on) and mixed noise. [ABSTRACT FROM PUBLISHER]