> dimensions to an arbitrary dimensionality is not a typical dimensionality reduction technique so that likely requires a special training/alignment technique that's novel

Very basic techniques (e.g. SuperBit random projection) have been extremely effective with OpenAI embeddings in the past. E.g. all embeddings on findsight.ai are OpenAI Ada embeddings stored as SuperBit signatures with a code length of 10,000 (i.e. 157 integers each), and there's almost no recall loss compared to the full vectors.