TF-IDF and BM25 Are Exact KL Divergences

We propose Latent Terms, a method revealing that models trained for dense retrieval, whether single- or multi-vector, learn representations that can trivially be decomposed into retrieval-ready sparse features. When trained on frozen retrievers, Sparse Autoencoders without any retrieval-specific adjustments extract a latent vocabulary with approximately Zipfian collection statistics, directly suitable for classical sparse retrieval scoring via BM25. This approach enables sparse retrieval while requiring no learned expansion objective or sparse retrieval supervision whatsoever, and can be readily applied to any dense retriever. Latent Terms is able to match or outperform single-vector scoring methods from its own base model as well as comparable SPLADE variants. In addition, it substantially outperforms its base model on LIMIT, a task specifically designed to highlight the failures of single-vector retrieval. Overall, our results highlight that neural retrievers contain more expressive and indexable structure than their default scoring functions expose, but that other methods can nonetheless be leveraged.

In retrieval-augmented coding, failures often begin when the relevant file is absent from the retrieved context. Under frozen generic tokenization, where a BM25 index has been built by a search system whose analyzer the practitioner does not control, this failure is routine: BM25's logarithmic RSJ-odds IDF under-separates the identifier tail that distinguishes one function from another. We replace the outer logarithm of the Robertson-Spärck-Jones odds with a q-logarithm. At q=1 the transform recovers BM25 exactly by L'Hôpital's rule, and for q<1 it is a Box-Cox transform of the RSJ odds with lambda = 1-q. On CoIR CodeSearchNet Go (182K documents), oracle-tuned NDCG@10 rises from 0.2575 to 0.4874 (absolute +0.2299; +89.3% relative; zero sign reversals in 10,000 paired-bootstrap resamples, reported as p <= 10^-4). The effect is graded across code languages and is near-zero on BEIR text. A one-parameter closed form estimates a corpus-level q from hapax density and stays near q=1 on corpora where BM25 is already optimal. The index-time cost is a single pass over the sparse score matrix and query latency is unchanged. A tokenizer ablation shows that identifier-aware tokenization largely removes the incremental gain from q-IDF.

TF-IDF is a classical formula that is widely used for identifying important terms within documents. We show that TF-IDF-like scores arise naturally from the test statistic of a penalized likelihood-ratio test setup capturing word burstiness (also known as word over-dispersion). In our framework, the alternative hypothesis captures word burstiness by modeling a collection of documents according to a family of beta-binomial distributions with a gamma penalty term on the precision parameter. In contrast, the null hypothesis assumes that words are binomially distributed in collection documents, a modeling approach that fails to account for word burstiness. We find that a term-weighting scheme given rise to by this test statistic performs comparably to TF-IDF on document classification tasks. This paper provides insights into TF-IDF from a statistical perspective and underscores the potential of hypothesis testing frameworks for advancing term-weighting scheme development.