How to decode from scikit-learn embedding?

Question

If I have a data matrix X, in which I want to learn a manifold embedding:

from sklearn.manifold import MDS

mds = MDS()
embedding = mds.fit_transform(X)

I can get back a 2D embedding/encoding of the original data X in the variable embedding.

Is there a way to "decode"/de-embed a given 2D point back to the original data dimension?

Answer 1

99% of embeddings used in ML are not injective thus there is no such thing as an inverse transformation (it is not even about being hard, it literally cannot exists as it transforms huge chunks of the space to a single point). In particular, MDS is not injective thus there is no way to go back .