An $$l_1$$ l 1 -norm loss based twin support vector regression and its geometric extension

Abstract

This paper proposes a novel $$l_1$$ -norm loss based twin support vector regression ( $$l_1$$ -TSVR) model. The bound functions in this $$l_1$$ -TSVR are optimized by simultaneously minimizing the $$l_1$$ -norm based fitting and one-side $$\epsilon$$ -insensitive losses, which results in different dual problems compared with twin support vector regression (TSVR) and $$\epsilon$$ -TSVR. The main advantages of this $$l_1$$ -TSVR are: First, it does not need to inverse any kernel matrix in dual problems, indicating that it not only can be optimized efficiently, but also has partly sparse bound functions. Second, it has a perfect and practical geometric interpretation. In the spirit of its geometric interpretation, this paper further presents a nearest-points based $$l_1$$ -TSVR (NP- $$l_1$$ -TSVR), in which bound functions are constructed by finding the nearest points between the reduced convex/affine hulls of training data and its shifted sets, respectively. Computational results obtained on a number of synthetic and real-world benchmark datasets clearly illustrate the superiority of the proposed $$l_1$$ -TSVR and NP- $$l_1$$ -TSVR as comparable generalization performance is achieved in accordance with the other SVR-type algorithms.