Learning with Euler Collaborative Representation for Robust Pattern Analysis

The Collaborative Representation (CR) framework has provided various effective and efficient solutions to pattern analysis. By leveraging between discriminative coefficient coding (l₂ 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.