A novel $ \ell_{0} $ minimization framework of tensor tubal rank and its multi-dimensional image completion application

Recently, minimizing the tensor tubal rank based on the tensor singular value decomposition (t-SVD) has attracted significant attention in the tensor completion task. The widely-used solutions of tensor-tubal-rank minimization rely upon various convex and nonconvex surrogates of the tensor rank. However, these tensor rank surrogates usually lead to inaccurate descriptions of the tensor rank. To mitigate the limitation, we propose an innovative $ \ell_0 $ minimization framework with guaranteed convergence to provide a novel paradigm for minimization of the tensor rank. To demonstrate the effectiveness of our framework, we develop a new tensor completion model employing a tensor adaptive sparsity-deduced rank (TASR). Subsequently, we formulate an algorithm rooted in the proposed $ \ell_0 $ minimization framework to address this model effectively. Experimental results on multi-dimensional image data demonstrate that our method is superior to several state-of-the-art approaches. The code is accessible at https://github.com/Jin-liangXiao/L0-TC.