Subspace clustering aims to seek a multi-subspace representation that is best suitable for data points taken from a high-dimensional space. Sparse representation and low-rank approximation-based methods have become one of the main melodies for subspace clustering. In the existing methods, nuclear norm is used to approximate rank minimization. However, the common deficiency still exists for nuclear norm, which always over-penalizes large singular values and results in a biased solution. In this paper, we propose a nonlinear subspace clustering model that exploits sparsity and low-rank of data in high dimensional feature space by using Schatten-[Formula: see text] norm surrogate ([Formula: see text]) with learned low-rank kernel. By this manner, the model guarantees that the data mapped in the high-dimensional feature spaces is lower rank and self-expressive. And we show the alternating direction method of multipliers (abbreviated as ADMM) for the corresponding problem in a reproducing kernel Hilbert space. Various experiments on motion segmentation and image clustering display that the proposed model has potentiality in outperforming most of state-of-the-art models in current literature.
Image completion, which falls to a special type of inverse problems, is an important but challenging task. The difficulties lie in that (i) the datasets usually appear to be multi-dimensional; (ii) the unavailable or corrupted data entries are randomly distributed. Recently, low-rank priors have gained importance in matrix completion problems and signal separation; however, due to the complexity of multi-dimensional data, using a low-rank prior by itself is often insufficient to achieve desirable completion, which requires a more comprehensive approach. In this paper, different from current available approaches, we develop a new approach, called relative total variation (TRTV), under the tensor framework, to effectively integrate the local and global image information for data processing. Based on our proposed framework, a completion model embedded with TRTV and tensor p-shrinkage nuclear norm minimization with suitable regularization is established. An alternating direction method of multiplier (ADMM)-based algorithm under our framework is presented. Extensive experiments in terms of denoising and completion tasks demonstrate our proposed method are not only effective but also superior to existing approaches in the literature.
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