Shared or dual projections? A bias–variance boundary gives the criterion, and CARS cuts held-out regret by 49–96% across five datasets

The paper characterizes the operator geometry of the two projection families: shared projections induce positive-semidefinite operators, dual projections realize arbitrary low-rank operators, and it derives the exact approximation loss imposed by sharing. Prior comparisons of shared and asymmetric dual encoders in dense retrieval were largely empirical, lacking a theoretical criterion for when the extra asymmetric flexibility is worth its cost; this work writes that choice as an exact approximation gap between operator classes. Theorem 1 gives the exact shared approximation error, decomposed into skew energy, negative eigenvalues, and discarded positive eigenvalues, with a proof in Appendix A; the retrieval interpretation equates this gap to the gap in squared optimal rank positive–negative separation under a unit negative-score second-moment constraint after whitening.

In a local Gaussian model the paper proves a phase boundary: dual has lower risk exactly when its squared directional signal exceeds the estimation cost of its additional degrees of freedom. This turns the intuition that dual is more flexible but noisier into a computable threshold, and gives an explicit count of the dual-only tangent directions. Theorem 2 gives the local estimation cost and the number of dual-only directions, and Corollary 1 states the phase condition; Appendix B proves the tangent-space dimensions, the derivative of the nearest-point projection, and risk convergence, and explains that the result is local because the sets are stratified at rank-changing points.

The paper derives a Stein-unbiased selection rule and then introduces CARS, a cross-fitted selector for real embeddings. The plug-in estimate is biased upward, so the work applies an unbiased correction and subtracts sampling variation across disjoint training halves, making the rule usable when the population geometry and noise covariance are unknown. Theorem 3 gives exact selection power via a noncentral chi-squared variable and a model-choice regret formula; Appendix C.1 proves the expectation identity for the CARS score and states explicitly that it is an expectation calculation and, unlike the Gaussian SURE theorem, does not assert an exact finite-sample selection probability.

Theory-guided retrieval experiments show the dual advantage widening with query rotation, a preference shift from shared to dual as training data grow, and CARS outperforming both fixed geometries across five datasets. The experiments test the predicted signal and sample-size effects across controlled simulations, full-corpus retrieval, and held-out operator risk, rather than reporting a single benchmark comparison. The mean Dual-minus-Shared NDCG@10 advantage more than doubles as query rotation increases from 0 to 90 degrees; in the rank–sample-size grids Shared wins 13 of 16 cells at n=32 while Dual wins all 32 cells at n=1024 and n=2048; all 168 comparable operator-risk curves move toward Dual as training data grow; CARS wins 85 of 100 encoder–fold comparisons, reaches 90.1% mean selection accuracy, and reduces regret by 49–96% relative to the better fixed choice.

The framework targets low-rank bilinear scoring over frozen query and document embeddings, and applies to retrieval-adaptation settings where one must decide whether to adopt asymmetric projections, such as evidence retrieval in retrieval-augmented generation, semantic search, and question answering. The theoretical boundary assumes local, isotropic Gaussian operator noise, and the authors list extension to anisotropic noise and directly to ranking metrics as future directions. CARS is meant for settings with available training data where the preferred geometry varies with training size, rank, and dataset; the reported selection study uses five datasets, four base encoders, nested training sizes and several ranks, with 20 internal half-splits.

A gap remains between the local Gaussian setting of the theory and the noise structure of real embeddings, and the authors themselves list anisotropic noise and directly optimizing ranking metrics as open directions. The CARS expectation identity is an expectation-level calculation and does not guarantee an exact finite-sample selection probability, so its selection stability under small samples or strong distribution shift still needs observation in more settings. The rotation intervention in the retrieval experiments rotates only query vectors while document embeddings, evaluation corpora, and relevance labels stay fixed; this controlled design helps isolate the signal effect, but its relation to natural distribution shift remains open. Appendix H also notes that a bilinear operator does not determine projected dual cosine, so operator-level conclusions require bilinear scoring or additional norm control.