In the following slide I do not understand the definition of the term embedding.
In the third paragraph, it says it is a mapping from low-dim. to high-dim, but in the last paragraph it suggests that it is a map from high-dim. to low-dim. (which sounds quite natural to me).
Is the definition (third paragraph) incorrect? 
Fortunately, I found on these slides an example:
The mapping $x = W z$ defines an embedding of an $m$-dimensional manifold in $p$-dimensional space
where $x$ is the original data. Moreover, now I understand why it is like this: the low-dim. space is embedded into the high-dim. space though an embedding (given a low-dim. vector, one applies the embedding to obtain the embedded version of the low-dim. vector into the high-dim. space).
While learning about a concept , never go with only a single source , try and cross validate with different texts , to confirm any hypothesis or definition.
This snippet itself is not comprehensive enough , and there should be more text supporting as to , what exactly embedding of manifold really is , and how well it should be understood to distinguish the so called "method" in this text.
Generally , embedding methods even in non-linear , or higher dimension can be visualized as regularization or decision trees.
