Current studies have demonstrated that the representational power of predictive state representations (PSRs) is at least equal to the one of partially observable Markov decision processes (POMDPs). This is while early steps in planning and generalization with PSRs suggest substantial improvements compared to POMDPs. However, lack of practical algorithms for learning these representations severely restricts their applicability. The computational inefficiency of exact PSR learning methods naturally leads to the exploration of various approximation methods that can provide a good set of core tests through less computational effort. In this paper, we address this problem in an optimization framework. In particular, our approach aims to minimize the potential error that may be caused by missing a number of core tests. We provide analysis of the error caused by this compression and present an empirical evaluation illustrating the performance of this approach.