Kashin Frames as Information-Preserving Representations for Retrieval

Dense retrieval for Retrieval-Augmented Generation (RAG) stores one float vector per passage, so index memory grows linearly with the corpus and quickly dominates serving cost at scale. Binary codes cut that footprint, but naive sign quantization of standard transformer embeddings loses ranking quality because a few heavy coordinates dominate similarity. We study Kashin frames as a posthoc, codebook-free low-bit representation for fixed encoders: map each embedding into a slightly overcomplete tight frame, clip coefficients to a Kashin bound, then pack 1-bit or 2-bit codes. This is an alternative to quantization-aware training (e.g., Matryoshka / binary-tuned embedders) when the model cannot be retrained. Frame redundancy is the operator dial between stored bytes and ranking retention. On BEIR SciFact, NFCorpus, and ArguAna with sentence-transformers/all-MiniLM-L6-v2 (???? = 384), Kashin 2-bit keeps 97.1% of FP32 nDCG@10 at 144 bytes (10.7×), and Kashin 1-bit keeps 92.5% at 72 bytes (21.3×). Both beat naive 1bit (89.2% at 48 bytes). At the same budgets, FAISS PQ-72 retains 97.5% and OPQ-144 retains 99.8%: trained codebooks win matchedbyte quality. Product-quantizing the Kashin coefficients instead of the raw embedding helps OPQ at 72 bytes (98.3% vs 96.9%) but not plain PQ. Bounding dynamic range before binarization remains a practical lever for compact indexes over legacy or black-box embeddings.