Before revision: The ability to preserve local geometry of highly nonlinear manifolds in high dimensional spaces and properly unfold them into lower dimensional hyperplanes is the key to the success of manifold computing, nonlinear dimensionality reduction (NLDR) and visualization.
After revision: The ability to preserve local geometry of highly nonlinear manifolds in high dimensional spaces and properly unfold them into lower dimensional hyperplanes is the key to the success of manifold computing, nonlinear dimensionality reduction (NLDR) plays an important role in feature extraction and visualization of high dimensional data. It transforms patterns of interest, or manifolds, in data to regions in a lower-dimensional latent space. When the manifolds are highly complicated, the NLDR transformation has to provide sufficient nonlinearity for downstream tasks.