arXiv:2601.20844v3 Announce Type: replace Abstract: This paper studies the Minimal Embeddable Dimension (MED): the least dimension in which there exists a configuration of $m$ object vectors so that every subset of size at most $k$ is exactly retrieved by score comparison. Our result shows MED is $\Theta(k)$, independent of $m$, for inner product, Euclidean distance, and cosine similarity. We then consider Robust MED (RMED), where all vectors are unit normed and an $\epsilon$ gap of scores is required. We derive the $m$-dependent feasibility ceiling $\epsilon_\star(m,k)=m/\sqrt{k(m-1)(m-k)}$,
The paper provides a theoretical underpinning for efficient retrieval systems at a time when embedding-based search and generative AI are rapidly advancing.
This research provides mathematical limits and optimal dimensions for embedding spaces, which are crucial for scaling AI applications reliant on similarity search, impacting efficiency and resource allocation.
Understanding the minimal embeddable dimension (MED) and robust MED (RMED) allows for more theoretically sound, potentially less resource-intensive, and more performant embedding models for retrieval.
- · AI model developers
- · Search engine companies
- · Cloud providers (via optimization)
- · Academia (theoretical foundations)
- · Inefficient embedding models
This research directly refines how embedding spaces are designed and optimized for retrieval tasks.
Improved embedding efficiency could lead to more performant and cost-effective AI applications across various domains, accelerating adoption.
As AI systems become more adept at understanding and retrieving complex information, this could indirectly contribute to the development of more sophisticated AI agents and knowledge systems.
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Read at arXiv cs.LG