Not really. See the literature on sharing lm_head (last matrix multiplication) with the input embedding dict.
Basically, the lm_head (a MxN matrix where M is the dictionary size and N is the internal dimension) can be seen as the dictionary too. You can think that and the softmax over it as compute cosine similarity of the last hidden output w.r.t. input embedding dictionary.
In that sense, they are sharing the representation space.
(BTW, I believe sharing lm_head with input embedding not working as good as separating them, so only mobile focused LLMs do so. So here is that. It would be interesting to experiment if injecting a projection layer like you suggested would improve performance or just red-herring).
Basically, the lm_head (a MxN matrix where M is the dictionary size and N is the internal dimension) can be seen as the dictionary too. You can think that and the softmax over it as compute cosine similarity of the last hidden output w.r.t. input embedding dictionary.
In that sense, they are sharing the representation space.
(BTW, I believe sharing lm_head with input embedding not working as good as separating them, so only mobile focused LLMs do so. So here is that. It would be interesting to experiment if injecting a projection layer like you suggested would improve performance or just red-herring).