When Does Dense Retrieval Need Asymmetric Geometry? A Bias-Variance Theory of Shared and Dual Projections

2026-09-26

AI Summary

This study addresses the lack of theoretical justification for selecting between shared and dual projection geometries in dense retrieval. We establish a bias-variance theoretical framework for low-rank bilinear scoring, deriving approximation gaps and Gaussian bounds to theoretically characterize, for the first time, the conditions under which dual projections are advantageous. Building upon this analysis, we propose CARS, a method that introduces a cross-validated asymmetric risk selector to dynamically and adaptively optimize retrieval geometries. Experimental results demonstrate that the benefits of dual projections become increasingly pronounced as data volume grows. Furthermore, CARS reduces leave-one-out regret by 49%–96% and achieves a geometry selection accuracy of 90.1%, validating the effectiveness of our theoretically grounded approach.

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.

Problem

• Dense Retrieval
• Shared Projections
• Dual Projections
• Bias-Variance Theory
• Geometry Selection

Innovation

• Dense Retrieval
• Bias-Variance Theory
• Dual Projections
• CARS
• Low-rank Bilinear Scoring