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(IDF)k=⌈log⁡2n⌉−⌈log⁡2dk⌉+1(\mathrm{IDF})_k = \lceil \log_2 n \rceil - \lceil \log_2 d_k \rceil + 1

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Before any of this involved learning, it involved geometry. Salton, Wong, and Yang proposed representing each document as a vector of weighted index terms and ranking documents by the similarity of their vectors to a query vector, arguing that a well-separated document space — one where unrelated documents sit far apart — should correspond to better retrieval performance than a densely packed one [ 1 ] . Their paper is worth reading in the original rather than through summary, because the term-weighting scheme it specifies is exactly the ancestor of what every later retriever, sparse or dense, still does: score a term by how often it occurs locally and how rare it is globally. They define…

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(IDF)k=⌈log⁡2n⌉−⌈log⁡2dk⌉+1,(\mathrm{IDF})_k = \lceil \log_2 n \rceil - \lceil \log_2 d_k \rceil + 1,

Equation 5 · AI Agents & Systems

From BM25 to Agentic Retrieval: A History of Retrieval-Augmented Generation

This equation states an equality: the expressions on both sides have the same value under the article’s assumptions.

Before any of this involved learning, it involved geometry. Salton, Wong, and Yang proposed representing each document as a vector of weighted index terms and ranking documents by the similarity of their vectors to a query vector, arguing that a well-separated document space — one where unrelated documents sit far apart — should correspond to better retrieval performance than a densely packed one [ 1 ] . Their paper is worth reading in the original rather than through summary, because the term-weighting scheme it specifies is exactly the ancestor of what every later retriever, sparse or dense, still does: score a term by how often it occurs locally and how rare it is globally. They define…

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