A generalized adaptive robust distance metric driven smooth regularization learning framework for pattern recognition

•A novel generalized adaptive robust distance metric is presented, called Lδ(u).•The Lδ(u) has some desirable salient properties, such as symmetry, boundedness, robustness, nonconvexity, and adaptivity, with both first-order and higher-order moments from samples.•The Lδ(u) can pick different robust distance metrics for different learning tasks during the learning process by the adaptive parameter δ.•A smooth regularized TELM learning framework for supervised and semi-supervised classification is proposed by introducing Lδ(u), regularization and smoothing techniques.•The fast Newton-Armijo algorithm and DC (difference of convex functions) programming algorithm (DCA) are used to solve the proposed methods. In this work, we propose a novel generalized adaptive robust distance metric called Lδ(u). Compared with other distance metrics, Lδ(u) has some desirable salient properties, such as symmetry, boundedness, robustness, nonconvexity, and adaptivity, with both first-order and higher-order moments from samples. On the other hand, Lδ(u) can pick different robust distance metrics for different learning tasks during the learning process by the adaptive parameter δ. Furthermore, we apply Lδ(u) to twin extreme learning machine (TELM) and develop an smooth regularized TELM learning framework for supervised and semi-supervised classification. By introducing the structural risk minimization (SRM) principle and smoothing techniques, the learning framework perfectly overcomes the computational burden associated with the matrix inversion operation required during TELM solving, while also significantly improving performance. More importantly, the proposed learning framework not only improves the robustness of TELM, but also can effectively use the geometric information of the marginal distribution embedded in the unlabeled samples to construct a more reasonable classifier. Finally, the globally convergent and quadratic convergent fast Newton-Armijo algorithm and DC (difference of convex functions) programming algorithm (DCA) are designed to solve the proposed methods. Experimental results demonstrate the effectiveness and robustness of our methods.