Transition state theory is a standard framework for predicting the rate of a chemical reaction. Although the transition state theory has been successfully applied to numerous chemical reaction analyses, many experimental and theoretical studies have reported chemical reactions with a reactivity which cannot be explained by the transition state theory due to dynamic effects. Dynamical systems theory provides a theoretical framework for elucidating dynamical mechanisms of such chemical reactions. In particular, reactive islands are essential phase space structures revealing dynamical reaction patterns. However, the numerical computation of reactive islands in a reaction system of many degrees of freedom involves an intrinsic challenge -- the curse of dimensionality. In this paper, we propose a dimensionality reduction algorithm for computing reactive islands in a reaction system of many degrees of freedom. Using the supervised principal component analysis, the proposed algorithm projects reactive islands into a low-dimensional phase space with preserving the dynamical information on reactivity as much as possible. The effectiveness of the proposed algorithm is examined by numerical experiments for H\'enon-Heiles systems extended to many degrees of freedom. The numerical results indicate that our proposed algorithm is effective in terms of the quality of reactivity prediction and the clearness of the boundaries of projected reactive islands. The proposed algorithm is a promising elemental technology for practical applications of dynamical systems analysis to real chemical systems.
