What Limits Does Quantization Place on Dense Top-$k$ Retrieval? A Theoretical Study

A theoretical study proves that quantization fundamentally limits dense top-k retrieval systems, requiring embedding dimension and precision to scale logarithmically with corpus size, contradicting prior corpus-independent bounds that assumed infinite precision. This finding has direct implications for practical vector databases and dense retrieval systems where quantization is standard practice.

This theoretical computer science paper addresses a critical gap between idealized retrieval models and real-world implementations. Previous research established that top-k retrieval could work with embedding dimensions scaling only with k (the number of retrieved items), regardless of corpus size N. However, this assumed infinite numerical precision—an unrealistic assumption for deployed systems.

The authors rigorously prove that with B bits per coordinate, achieving perfect top-k retrieval demands Bd = Ω(k ln N), meaning the total bit budget must grow logarithmically with corpus size. They further identify a precision threshold B* = O(ln ln N) below which no dimension can compensate, establishing three distinct regimes for feasible (B, d) parameter pairs under ℓ2-normalized uniform scalar quantization.

For the vector database and dense retrieval industry, this work validates a longstanding engineering intuition: you cannot arbitrarily compress embeddings without consequences. Companies like Pinecone, Weaviate, and Milvus—which optimize quantization for production systems—must contend with this fundamental tradeoff. Smaller embedding dimensions and lower precision reduce storage and computational costs, but the theoretical bounds show these optimizations cannot scale indefinitely across arbitrarily large corpora.

The implications extend to AI infrastructure generally. As organizations build retrieval-augmented generation systems for massive datasets, they face a mathematical constraint: either accept growing embedding dimensions, increase precision costs, or accept degraded retrieval quality. This research provides the theoretical foundation for understanding where existing systems approach fundamental limits.

• →Quantization introduces fundamental limits: embedding dimension must grow logarithmically with corpus size, contradicting prior corpus-independent bounds.
• →A precision threshold exists below which no embedding dimension can achieve perfect top-k retrieval, establishing hard limits for ultra-low-bit quantization.
• →Vector databases cannot indefinitely scale by reducing precision and dimension; tradeoffs follow mathematically bounded regimes.
• →The theoretical gap between infinite-precision models and practical B-bit systems has real consequences for production retrieval systems.
• →Organizations deploying dense retrieval at scale must balance embedding dimension, quantization precision, and corpus size within these proven constraints.