This paper presents a new variable selection approach integrated with\nGaussian process (GP) regression. We consider a sparse projection of input\nvariables and a general stationary covariance model that depends on the\nEuclidean distance between the projected features. The sparse projection matrix\nis considered as an unknown parameter. We propose a forward stagewise approach\nwith embedded gradient descent steps to co-optimize the parameter with other\ncovariance parameters based on the maximization of a non-convex marginal\nlikelihood function with a concave sparsity penalty, and some convergence\nproperties of the algorithm are provided. The proposed model covers a broader\nclass of stationary covariance functions than the existing automatic relevance\ndetermination approaches, and the solution approach is more computationally\nfeasible than the existing MCMC sampling procedures for the automatic relevance\nparameter estimation with a sparsity prior. The approach is evaluated for a\nlarge number of simulated scenarios. The choice of tuning parameters and the\naccuracy of the parameter estimation are evaluated with the simulation study.\nIn the comparison to some chosen benchmark approaches, the proposed approach\nhas provided a better accuracy in the variable selection. It is applied to an\nimportant problem of identifying environmental factors that affect an\natmospheric corrosion of metal alloys.\n