What exactly is called "embedding" in dimensionality reduction?

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?

• I can't see your image/slide. Sep 10 '18 at 10:23
• Weird, I see it. It can also be found here: i.stack.imgur.com/vT05O.png Sep 10 '18 at 10:31
• The last paragraph clearly states it is discussing situations where the method does not create a manifold and so no embedding either. There is no conflict Sep 13 '18 at 7:56
• Perhaps you are right. I was disturbed by the word "but". I interpreted that sentence as: they create a manifold, but they do not create an embedding. Sep 13 '18 at 8:14

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).