Reversing linear dimensionality reduction algorithms by projecting new set of data in the learned manifold is very straight forward. I am interested in projecting new set of data on non-linear manifolds (eg.Sammon Mapping) and then reconstruct the high dimensional data using specific amount of components. Is there a way of reversing non-linear dimensionality reduction alorithms?
Some nonlinear dimensionality reduction (NLDR) methods provide an explicit mapping from the low dimensional space back to the original, high dimensional space. But, most don't. Autoencoders intrinsically give this capability. An inverse mapping can be approximated for locally linear embedding (LLE) using a form of local regression (similar to what LLE uses to compute the low dimensional embedding in the first place). For other methods, one can try to learn the inverse mapping by treating it as a regression problem: train a model to predict the high dimensional points, given the low dimensional points.