Statistical machine learning, such as kernel methods, have been widely used to discover hidden regularities and patterns in data. In particular, one-class classification algorithms gained a lot of interest in a large number of applications where the only available data designate a unique class, as in industrial processes. In this paper, we propose a sparse framework for one-class classification problems, by investigating the hypersphere enclosing the samples in a Reproducing Kernel Hilbert Space (RKHS). The center of this hypersphere is approximated using a sparse solution, by selecting an appropriate set of relevant samples. For this purpose, we investigate well-known shrinkage methods, namely Least Angle Regression, Least Absolute Shrinkage and Selection Operator, and Elastic Net. We revisit these methods and adapt their algorithms for estimating the sparse center in the RKHS. The proposed framework is extended to include the truncated Mahalanobis distance, which is necessary when dealing with heterogenous input variables. We also provide some theoretical results on the projection error and on the error of the first kind. The proposed algorithms are compared with well-known one-class classification approaches, with experiments conducted on simulated and real datasets.