PCA cannot take into account nonlinear structures, structures consisting of arbitrarily shaped clusters or curved manifolds, since it describes the data in terms of a linear subspace. Projection pursuit tries to express some nonlinearities, but if the data set is high-dimensional and highly nonlinear it may be difficult to visualize it with linear projections onto a low-dimensional display even if the ``projection angle'' is chosen carefully.
Several approaches have been proposed for reproducing nonlinear higher-dimensional structures on a lower-dimensional display. The most common methods allocate a representation for each data point in the lower-dimensional space and try to optimize these representations so that the distances between them would be as similar as possible to the original distances of the corresponding data items. The methods differ in how the different distances are weighted and how the representations are optimized.