Adapting Product Quantization to Hierarchical MRL/CSRv2 Embeddings

Develop a product quantization approach that is adapted to the hierarchical representations produced by Matryoshka Representation Learning and CSRv2, so that quantization functions effectively under fixed bit budgets for these prefix-concentrated embedding structures.

Background

The authors observe that standard product quantization (PQ) underperforms binary quantization for MRL/CSRv2 due to a structural mismatch: PQ allocates bits uniformly across subspaces, while MRL/CSRv2 concentrate core semantics in early dimensions, yielding hierarchical embeddings.

They explicitly state that adapting PQ to work effectively with this hierarchy is left for future work, framing a concrete next step for efficient compression of such embeddings.

References

However, as discussed in Appendix \ref{appendix:fixed_memory_cost}, while PQ presents an interesting avenue, it necessitates adaptation to function effectively with MRL/CSRv2's hierarchical representations, which we leave for future work.

— CSRv2: Unlocking Ultra-Sparse Embeddings  (2602.05735 - Guo et al., 5 Feb 2026) in Appendix: Potential Applications of CSRv2 in Vector Quantization

However, it remains to show whether a concrete mathematical or geometric description of the embeddings can be derived from the loss function.

— Quantization Beyond Uniform Bit Allocation  (2608.19388 - Sreeramji et al., 19 Aug 2026) in Section 5, “Limitations and Future Work,” item 1