Dimension reduction plays an important role in the community of high dimensional data analysis. The notion of random anisotropic transform (RAT), which was applied to speed up the computation procedure of dimension reduction kernel (DRK) with Isomap embedding (Isomap-RAT), was introduced in this paper. Nevertheless, traditional Isomap-RAT does not consider the intrinsic dimension that the hyperspectral image data resides on. Moreover, The DRK of Isomap embedding is not always guaranteed to be positive semi-definite. Thus, this paper proposed a kernel Isomap-Hysime random anisotropic transform (KIH-RAT) to deal with these challenges that met frequently in reality. The proposed methodology consists of two main terms: 1) a kernel term that finds an approximative constant which is added to the dissimilar matrix to make the DRK to be positive semi-definite; and 2) an intrinsic dimension assessment term that employs Hysime to estimate the intrinsic dimension of hyperspectral image data to preserve the geometries of original information as much as possible. The proposed method is exhaustively tested on two reduced feature spaces that relate to the classification of real hyperspectral remote sensing images. The effectiveness and feasibility of presented KIH-RAT methodology are illustrated by the experiment results from both real hyperspectral image examples.