We propose a new approach to analyze data that naturally lie on manifolds. We
focus on a special class of manifolds, called direct product manifolds, whose
intrinsic dimension could be very high. Our method finds a low-dimensional
representation of the manifold that can be used to find and visualize the
principal modes of variation of the data, as Principal Component Analysis (PCA)
does in linear spaces. The proposed method improves upon earlier manifold
extensions of PCA by more concisely capturing important nonlinear modes.