When Does Dense Retrieval Need Asymmetric Geometry? A Bias-Variance Theory of Shared and Dual Projections
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 ...
| Authors | Maojun Sun, Yancheng Yuan, Jian Huang et al. |
|---|---|
| Primary category | cs.IR |
| Published | 26 September 2026 |
| Last updated at source | 26 September 2026 |
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