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

When Does Dense Retrieval Need Asymmetric Geometry? A Bias-Variance Theory of Shared and Dual Projections. By Maojun Sun and three others, from the Hong Kong Polytechnic University. Published on the twenty-sixth of September. No code repository is linked to it.

Dense retrieval sits underneath retrieval-augmented generation, semantic search, and question answering. One basic design choice is whether queries and documents pass through a shared projection or through two separate ones, and the theoretical basis for choosing has been unclear. The authors study low-rank bilinear scoring, and build a bias-variance theory for exactly that choice.

The two options differ in what they can express. Shared projections induce positive-semidefinite operators. Dual projections can realize arbitrary low-rank operators. The authors derive the exact approximation gap between them, and prove a local Gaussian boundary: dual has lower risk exactly when the squared directional signal exceeds the estimation cost of its additional degrees of freedom.

Put plainly, dual buys expressiveness and pays for it in estimation. When the directional signal is strong, the extra freedom earns its keep. When it is weak, the extra parameters cost more than they give back. That boundary motivates a practical tool, the Cross-fitted Asymmetry Risk Selector, or CARS, which estimates reproducible directional signal from training pairs. Its Gaussian counterpart admits exact formulas for selection power and for regret.

The exact statements are local, Gaussian, and about low-rank bilinear scorers. To see how the picture holds up beyond those conditions, the authors run retrieval experiments across several datasets and embedding models. As query rotation increases from zero degrees to ninety, the mean NDCG at ten advantage of dual over shared more than doubles.

In grids over rank and sample size, shared wins thirteen of sixteen cells at thirty-two samples. Dual wins all thirty-two cells at one thousand and twenty-four samples, and again at two thousand and forty-eight. In the same direction, all one hundred sixty-eight comparable operator-risk curves move toward dual as training data grow. Against the two fixed-geometry baselines, CARS reduces held-out regret by between forty-nine and ninety-six percent, and selects the right geometry ninety point one percent of the time on average.

The answer to the title's question, then, is conditional. Dense retrieval needs asymmetric geometry when the directional signal is strong enough to pay for the extra freedom, and more training data tilts that balance toward dual. CARS turns the condition into a decision that can be made from the training pairs themselves.