Robustness to noises, outliers, and corruptions is an important issue in linear dimensionality reduction. Since the sample-specific corruptions and outliers exist, the class-special structure or the local geometric structure is destroyed, and thus, many existing methods, including the popular manifold learning- based linear dimensionality methods, fail to achieve good performance in recognition tasks. In this paper, we focus on the unsupervised robust linear dimensionality reduction on corrupted data by introducing the robust low-rank representation (LRR). Thus, a robust linear dimensionality reduction technique termed low-rank embedding (LRE) is proposed in this paper, which provides a robust image representation to uncover the potential relationship among the images to reduce the negative influence from the occlusion and corruption so as to enhance the algorithm's robustness in image feature extraction. LRE searches the optimal LRR and optimal subspace simultaneously. The model of LRE can be solved by alternatively iterating the argument Lagrangian multiplier method and the eigendecomposition. The theoretical analysis, including convergence analysis and computational complexity, of the algorithms is presented. Experiments on some well-known databases with different corruptions show that LRE is superior to the previous methods of feature extraction, and therefore, it indicates the robustness of the proposed method. The code of this paper can be downloaded from http://www.scholat.com/laizhihui.
Tensor-based object recognition has been widely studied in the past several years. This paper focuses on the issue of joint feature selection from the tensor data and proposes a novel method called joint tensor feature analysis (JTFA) for tensor feature extraction and recognition. In order to obtain a set of jointly sparse projections for tensor feature extraction, we define the modified within-class tensor scatter value and the modified between-class tensor scatter value for regression. The k-mode optimization technique and the L(2,1)-norm jointly sparse regression are combined together to compute the optimal solutions. The convergent analysis, computational complexity analysis and the essence of the proposed method/model are also presented. It is interesting to show that the proposed method is very similar to singular value decomposition on the scatter matrix but with sparsity constraint on the right singular value matrix or eigen-decomposition on the scatter matrix with sparse manner. Experimental results on some tensor datasets indicate that JTFA outperforms some well-known tensor feature extraction and selection algorithms.
Ridge regression is widely used in multiple variable data analysis. However, in very high-dimensional cases such as image feature extraction and recognition, conventional ridge regression or its extensions have the small-class problem, that is, the number of the projections obtained by ridge regression is limited by the number of the classes. In this paper, we proposed a novel method called generalized robust regression (GRR) for jointly sparse subspace learning which can address the problem. GRR not only imposes L 2,1 -norm penalty on both loss function and regularization term to guarantee the joint sparsity and the robustness to outliers for effective feature selection, but also utilizes L 2,1 -norm as the measurement to take the intrinsic local geometric structure of the data into consideration to improve the performance. Moreover, by incorporating the elastic factor on the loss function, GRR can enhance the robustness to obtain more projections for feature selection or classification. To obtain the optimal solution of GRR, an iterative algorithm was proposed and the convergence was also proved. Experiments on six wellknown data sets demonstrate the merits of the proposed method. The result indicates that GRR is a robust and efficient regression method for face recognition.
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