Shared or dual projections? A bias–variance boundary gives the criterion, and CARS cuts held-out regret by 49–96% across five datasets
The work develops a bias–variance theory for low-rank bilinear scoring in dense retrieval, proving that dual projections have lower risk exactly when squared directional signal exceeds the estimation cost of their extra degrees of freedom, and uses it to build the cross-fitted CARS selector, which reaches 90.1% mean geometry-selection accuracy and reduces held-out regret by 49–96% relative to the better fixed geometry across multiple datasets and embedding models.