The high dimensions of hyperspectral imagery have caused burden for further processing. A new Fast Independent Component Analysis (FastICA) approach to dimensionality reduction for hyperspectral imagery is presented. The virtual dimensionality is introduced to determine the number of dimensions needed to be preserved. Since there is no prioritization among independent components generated by the FastICA, the mixing matrix of FastICA is initialized by endmembers, which were extracted by using unsupervised maximum distance method. Minimum Noise Fraction (MNF) is used for preprocessing of original data, which can reduce the computational complexity of FastICA significantly. Finally, FastICA is performed on the selected principal components acquired by MNF to generate the expected independent components in accordance with the order of endmembers. Experimental results demonstrate that the proposed method outperforms second-order statistics-based transforms such as principle components analysis.