The Collaborative Representation (CR) framework has provided various effective and efficient solutions to pattern analysis. By leveraging between discriminative coefficient coding (l 2 regularization) and the best reconstruction quality (collaboration), the CR framework can exploit discriminative patterns efficiently in high-dimensional space. Due to the limitations of its linear representation mechanism, the CR must sacrifice its superior efficiency for capturing the non-linear information with the kernel trick. Besides this, even if the coding is indispensable, there is no mechanism designed to keep the CR free from inevitable noise brought by real-world information systems. In addition, the CR only emphasizes exploiting discriminative patterns on coefficients rather than on the reconstruction. To tackle the problems of primitive CR with a unified framework, in this article we propose the Euler Collaborative Representation (E-CR) framework. Inferred from the Euler formula, in the proposed method, we map the samples to a complex space to capture discriminative and non-linear information without the high-dimensional hidden kernel space. Based on the proposed E-CR framework, we form two specific classifiers: the Euler Collaborative Representation based Classifier (E-CRC) and the Euler Probabilistic Collaborative Representation based Classifier (E-PROCRC). Furthermore, we specifically designed a robust algorithm for E-CR (termed as R-E-CR ) to deal with the inevitable noises in real-world systems. Robust iterative algorithms have been specially designed for solving E-CRC and E-PROCRC. We correspondingly present a series of theoretical proofs to ensure the completeness of the theory for the proposed robust algorithms. We evaluated E-CR and R-E-CR with various experiments to show its competitive performance and efficiency.

In recent years, incomplete multi-view clustering has drawn increasing attention due to the existence of large amounts of unlabeled incomplete data whose views are not fully observed in the practical applications. Although many traditional methods have been extended to address the incomplete learning problem, most of them exploit the shallow models and ignore the geometric structure. To address these issues, we proposed a structural deep incomplete multi-view clustering network. Specifically, the proposed method can simultaneously explore the high-level features and high-order geometric structure information of data with several view-specific graph convolutional encoder networks and can directly obtain the optimal clustering indicator matrix in one stage. Experimental results on several datasets with the comparison of state-of-the-art methods validate the superiority of the proposed method.

Dictionary-based classification has been promising in knowledge discovery from image data, due to its good performance and interpretable theoretical system. Dictionary learning effectively supports both small- and large-scale datasets, while its robustness and performance depends on the atoms of the dictionary most of the time. Empirically, using a large number of atoms is helpful to obtain a robust classification, while robustness cannot be ensured when setting a small number of atoms. However, learning a huge dictionary dramatically slows down the speed of classification, which is especially worse on the large-scale datasets. To address the problem, we propose a Fast and Robust Dictionary-based Classification (FRDC) framework, which fully utilizes the learned dictionary for classification by staging - and -norms to obtain a robust sparse representation. The new objective function, on the one hand, introduces an additional -norm term upon the conventional -norm optimization, which generates a more robust classification. On the other hand, the optimization based on both - and -norms is solved in two stages, which is much easier and faster than current solutions. In this way, even when using a limited size of dictionary, which makes sure the classification runs very fast, it still can gain higher robustness for multiple types of image data. The optimization is then theoretically analyzed in a new formulation, close but distinct to elastic-net, to prove it is crucial to improve the performance under the premise of robustness. According to our extensive experiments conducted on four image datasets for face and object classification, FRDC keeps generating a robust classification no matter whether using a small or large number of atoms. This guarantees a fast and robust dictionary-based image classification. Furthermore, when simply using deep features extracted via some popular pre-trained neural networks, it outperforms many state-of-the-art methods on the specific datasets.