Dense retrieval accuracy depends on query and data size geometry choice
When Does Dense Retrieval Need Asymmetric Geometry? A Bias-Variance Theory of Shared and Dual Projections
Information RetrievalMachine Learning
Summary
Choosing how to match questions and documents in search systems can use either shared or separate projection methods. The authors explain when each method works better by balancing a trade-off between bias and variance. They introduce a selection method named CARS that picks the best approach based on training data signals. Their experiments show that as training data grows or queries become more complex, using separate projections often improves retrieval quality.
What this means in practice
- •For search engine developers: Optimize retrieval model geometry to improve relevance based on query complexity and training set size by using the CARS selection method.
- •For question answering system teams: Enhance answer retrieval accuracy by dynamically selecting between shared and dual projection embeddings depending on available training data.
Authors
Maojun Sun, Yancheng Yuan, Jian Huang, Ruijian Han
Abstract
Dense retrieval powers retrieval-augmented generation, semantic search, and question answering, yet the theoretical basis for choosing between shared and dual query-document projections remains unclear. We introduce a bias-variance theory for low-rank bilinear scoring. Shared projections induce positive-semidefinite operators, whereas dual projections realize arbitrary low-rank operators. We derive their exact approximation gap and prove a local Gaussian boundary: dual has lower risk exactly when squared directional signal exceeds the estimation cost of its additional degrees of freedom. This boundary motivates the Cross-fitted Asymmetry Risk Selector (CARS), which estimates reproducible directional signal from training pairs; its Gaussian counterpart admits exact selection-power and regret formulas. Guided by the theory, we run retrieval experiments across multiple datasets and embedding models. The mean Dual-minus-Shared NDCG@10 advantage more than doubles as query rotation increases from 0 degrees to 90 degrees. In the rank-sample-size grids, Shared wins 13 of 16 cells at n=32, whereas Dual wins all 32 cells at n=1024 and n=2048. Consistent with this shift, all 168 comparable operator-risk curves move toward Dual as training data grow. Compared to the two fixed-geometry baselines, CARS reduces held-out regret by 49-96% and achieves 90.1% mean geometry-selection accuracy.