Necessity of linear embedding dimension for dual encoders in retrieval

Determine whether the embedding dimension of dual encoders must grow linearly with the number of relevant documents in order to correctly separate relevant documents from irrelevant ones in retrieval tasks that do not consider ranking.

Background

The paper notes that prior work (Guo et al., 2019) established that a dual encoder (DE) embedding dimension growing linearly with the number of relevant documents is sufficient to separate relevant items from irrelevant ones in retrieval. However, that result concerns retrieval without ranking. The authors highlight that whether such linear growth is also necessary in the retrieval setting remains unresolved. They then proceed to prove necessity for ranking tasks, but do not close the question for retrieval separation.