Surrogate construction for high-dimensional models is challenged in two major ways: obtaining sufficient training model simulations becomes prohibitively expensive, and non-adaptive basis selection rules lead to excessively large basis sets. We enhanced select state-of-the-art tools from statistical learning to build efficient sparse surrogate representations, with quantified uncertainty, for high-dimensional complex models. Specifically, Bayesian compressive sensing techniques are supplemented by iterative basis growth and weighted regularization. Application to a 70-dimensional climate land model shows promising results.