Mapping one embedding to another using Deep Learning
You should take into account a few factors:


some_value is key here. You should probably choose a value greater than e1_dim and e2_dim. This may be a problem due to the large memory needs for high embedding dimensionalities.
if e2_dim is bounded to $(-1,+1)$, you may want to set a tanh activation in the last layer (i.e. after Dense(e2_dim)). With that in place, you may want to replace the activation of the hidden layer with a ReLU, to avoid squashing the gradients unnecessarily.
If your embeddings are directional, or they are in the unit ball, then you should be fine with cosine_proximity. Maybe a better option would be to use the von Mises-Fischer distance, but I think it's not straightforward. Otherwise, you may try World-Mover's Distance.


That being said, a lot of embedded space mapping works out there assume that the embedded spaces are approximately isomorphic and just go ahead with a linear transformation. You may want to have a look at the relevant bilingual embedding mapping literature; you may start with the work by Artetxe et al., 2016. I don't know if the mentioned assumption is met to any degree in your scenario (skipgram $\leftrightarrow$ node2vec), but you may assess this with the VF2 algorithm, like Artetxe et al. did.