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This study introduces a unified approach to tackle challenges in both low-level and high-level vision tasks for image processing. The framework integrates faster nonconvex low-rank matrix computations and continuity techniques to yield efficient and high-quality results. In addressing real-world image complexities like noise, variations, and missing data, the framework exploits the intrinsic low-rank structure of the data and incorporates specific residual measurements. The optimization problem for low-rank matrix learning is effectively solved using the nonconvex Proximal Block Coordinate Descent (PBCD) algorithm, resulting in nearly unbiased estimators. Rigorous theoretical analysis ensures both local and global convergence. The PBCD algorithm updates blocks of variables iteratively with closed-form solutions, adeptly handling nonconvexity and promoting faster convergence. Notably, the framework incorporates the randomized singular value decomposition (RSVD) technique and introduces a continuous strategy for adaptive model parameter updates. These strategic choices reduce computational complexity while maintaining result quality. They offer fine-tuned control over the desired rank of the learned matrix and enhance robustness in a straightforward manner. Furthermore, the versatility of the proposed nonconvex PBCD algorithm extends to handling problems with multiple variables, as supported by theoretical analysis. Experimental evaluations, spanning various image low-level and high-level vision tasks such as inpainting, classification, and clustering, validate the effectiveness and efficiency of our framework across diverse databases. The source code is available at https://github.com/ZhangHengMin/FNPBCD%5fLR. In a nutshell, our framework provides a unified solution to tackle both low-level and high-level vision tasks in images. By combining fast nonconvex low-rank matrix learning with adaptive parameter updates, we achieve efficient computation, yielding high-quality results that demonstrate robustness against various types of noise. The evaluations further endorse the reliability and applicability of our proposed framework. • Integrating nonconvex estimators serves to enhance learning capabilities; • Introducing a unified framework generalizes multiple rank-relaxed problems; • The combination of RSVD and continuity techniques yields an efficient solver; • The superiority is established and reinforced through experimental validations. [ABSTRACT FROM AUTHOR]

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]