# Parameterized Dense-Sparse Fusion for Hybrid Retrieval: Tuning a Rank-Score Mix on BEIR SciFact with Qdrant

arXiv: https://arxiv.org/abs/2609.22770 · HTML: https://mlsanity.com/paper/2609.22770/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 8 of its 15 references that are in the map.
- First posted on arXiv 19 Sep 2026.

## Builds on (most cited first, 8 of 8)

- [Sentence-BERT: Sentence Embeddings using Siamese BERT-Networks](https://mlsanity.com/paper/1908.10084/) (2019)
- [Dense Passage Retrieval for Open-Domain Question Answering](https://mlsanity.com/paper/2004.04906/) (2020)
- [BEIR: A Heterogenous Benchmark for Zero-shot Evaluation of Information Retrieval Models](https://mlsanity.com/paper/2104.08663/) (2021)
- [Passage Re-ranking with BERT](https://mlsanity.com/paper/1901.04085/) (2019)
- [MTEB: Massive Text Embedding Benchmark](https://mlsanity.com/paper/2210.07316/) (2022)
- [C-Pack: Packed Resources For General Chinese Embeddings](https://mlsanity.com/paper/2309.07597/) (2023)
- [SPLADE: Sparse Lexical and Expansion Model for First Stage Ranking](https://mlsanity.com/paper/2107.05720/) (2021)
- [Pretrained Transformers for Text Ranking: BERT and Beyond](https://mlsanity.com/paper/2010.06467/) (2020)

## Abstract (arXiv)

We study a parameterized hybrid ranker that fuses a dense embedding list and a sparse lexical list. The method has a small, explicit parameter vector: a dense prior α ∈ [0,1], a score-versus-rank mix λ ∈ [0,1], an RRF smoothing parameter kappa > 0, optional list-geometry coefficients that move α per query, and a router margin τ that can turn sparse search off. We grid-search those ranges on SciFact train (809 queries) and freeze the chosen values on SciFact test (300). The tuned rank-score mix (α = 0.8, λ = 0.75, kappa = 20) reaches 0.753 nDCG@10 and 0.889 recall@10, outperforming dense BGE (0.742 / 0.871) and equal-weight RRF (0.707 nDCG@10) on that test split. A list-conditioned α adds +0.0006 nDCG; a sparse-off router is rejected by the same train split (any τ that skipped approximately 50% of queries lost nDCG). These coefficients are dataset-specific. Equal RRF with the same models does not beat dense on a nine-zip BEIR macro-average (0.479 vs. 0.519 nDCG@10). Repeating the same train-then-freeze sweep independently on all 20 indexed units beats equal RRF on 20/20 and dense on 16/20 (unit-mean nDCG@10 0.467 vs. 0.462 dense vs. 0.420 RRF). Other corpora should reuse the ranges, not a copy of the SciFact point.