Word embedding and Euclidean distance

Does a transformation exist that allows to use of the Euclidean distance with the word embeddings? The Cosine distance could be a problem in my case. For example, what if I translate the vector to a polar coordinate system from the cartesian one? Does it make sense?

The idea is reduce the dimensionality of such vector using PCA or Autoencoders. In my understanding, using PCA on word embeddings means to produce random data.

How can I reduce the dimensionality of the word-embeddings saving the meaning the have?

• Regardless of coordinate system, distance is the same. Coordinate systems are just a representation of the same space. Commented Jun 30, 2023 at 0:54
• Yes, I agree. However, since we care about the direction rather than the module, reducing the vectors to be unitary does not imply information loss. At this point, only the angles matter. Further, yesterday I got a hint about that, which confirmed that in the case of unitary vectors, Euclidean distance matches the cosine distance. The first step is solved. About the meaningfulness of a dimensional reduced version of the embeddings... I'm afraid that it is not working. Commented Jun 30, 2023 at 13:54
• Both distance metrics are roughly the same stats.stackexchange.com/questions/544951/…
– Tim
Commented Jun 30, 2023 at 14:56
• @Tim, yes I agree. I had an intuition and the link you are referring is a consolidated known fact. Commented Jun 30, 2023 at 16:32