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.
Figures & tables
Figure 1: Shared and dual projections for dense retrieval. A shared map P induces a PSD operator M=P⊤P , while dual maps A and B induce a general operator M=A⊤B of rank at most r .
Figure 2: Geometry behind the boundary. A: the shared PSD family lies inside the dual family; δ schematically marks a target’s departure from sharing. B: locally, sharing removes k dual-only noise coordinates but also discards signal along them; k is the extra dimension derived in Section 3 .
Figure 3: A Shared-to-Dual crossover in controlled two-view retrieval. A: Mean Dual-minus-Shared NDCG@10 across mismatch and training size; the gold contour marks zero. B: The same gap versus population distance GPSD to the shared PSD family; bands show 95% normal confidence intervals.
Figure 4: Raw asymmetry is inflated, while correction better predicts held-out fit. A: Training PSD distance exceeds its population value; shading shows a 95% t interval. B: Held-out Shared-minus-Dual operator-loss advantage across raw and corrected score deciles. Positive values favor Dual.
Figure 5: Dual-minus-Shared full-corpus NDCG@10 under query rotation. Each bar averages three paired seeds for BGE-base and GTE-base on each dataset.
Figure 6: Rank–sample-size retrieval gaps. Each cell is mean Dual-minus-Shared test NDCG@10 over three paired seeds, in units of 10−4 ; blue favors Shared and red favors Dual.
Figure 7: Held-out operator fit moves toward Dual with more training queries. A: NQ; B: SciFact. Each point averages ranks and three seeds within one encoder; positive values indicate lower held-out loss for Dual.
Rule
Dataset
BGE-base
Contriever
E5-base-v2
GTE-base
Regret
Accuracy
Wins
Regret
Accuracy
Wins
Regret
Accuracy
Wins
Regret
Accuracy
Wins
Always Shared
ArguAna
0.0950
45.8%
0/5
0.1310
40.0%
0/5
0.2543
30.0%
0/5
0.1294
43.3%
0/5
MS MARCO
0.0680
75.6%
1/5
0.3254
46.2%
0/5
0.0839
75.0%
1/5
0.0442
79.4%
1/5
FEVER
0.2490
33.1%
1/5
0.4959
21.9%
0/5
0.3968
25.0%
0/5
0.2999
37.5%
1/5
NQ
0.0050
85.0%
1/5
0.1915
40.0%
0/5
0.1020
54.3%
1/5
0.0232
74.3%
2/5
SciFact
0.0226
63.3%
1/5
0.0288
58.3%
0/5
0.0202
65.0%
2/5
0.0374
65.0%
1/5
Table 1: Geometry selection on five datasets and four encoders. Lower held-out operator-risk regret and higher decision accuracy or strict outer-fold wins indicate better performance. Regret and accuracy are averaged over five held-out query folds.
Appendix figures & tables3 assets
Supplementary material from the paper’s appendix.
Appendix
δ2/(σ2k)
Shared risk
Dual risk
Pr(π=d)
0.0
32.97/33
63.03/63
0.001/0.001
0.5
48.05/48
63.05/63
0.094/0.093
1.0
62.93/63
62.97/63
0.473/0.474
2.0
93.10/93
63.15/63
0.970/0.969
Appendix
Table 2: Local risk and SURE calibration. Entries are simulation/theory; risks are scaled by n .
Figure 8: Held-out operator-risk trajectories for BGE-base and Contriever (84 curves). Rows are FEVER and NQ (24 curves each), followed by ArguAna and SciFact (18 each). Each line fixes a dataset, encoder, rank, and seed and connects measured nested training sizes. Color denotes rank; line style denotes seed. Positive normalized Shared-minus-Dual loss favors Dual.
Figure 9: The remaining 84 held-out operator-risk trajectories, for E5-base-v2 and GTE-base. Grouping, axes, colors, and line styles match Figure 8 .
Dense embedding retrieval compresses all relevance information into a single inner product, imposing a fundamental geometric limit -- the Voronoi Bottleneck -- on the number of query-document relevance patterns expressible at fixed embedding dimension (d). We make three contributions. (1) Unified capacity theory. We prove that Voronoi complexity and sign-rank are equivalent for top-1 retrieval, yielding tight dimension bounds and a computable diagnostic, the Capacity Utilization Score (CUS), that predicts per-query retrieval failure with AUC (> 0.8) without relevance labels. (2) Diagnosis. CUS identifies two capacity regimes -- moderate ((δ\gtrsim 1)), where density-aware training yields measurable gains, and vacuous ((δ\ll 1)), where it does not -- giving practitioners an a priori check before investing in retraining. (3) DART training. We introduce AT-DW-InfoNCE, an Adaptive-Temperature Density-Weighted contrastive objective with formally derived optimal weighting (α^* = 2.0). On a 100K-query synthetic product-search corpus with controlled relevance structure, DART improves +1.9 Recall@100 over a same-data InfoNCE baseline ((84.9 \pm 0.0) vs. (83.0 \pm 0.3); 8 seeds, (p < 0.001)), outperforming focal loss and temperature-schedule alternatives. DART requires zero inference-time overhead -- it is a drop-in training objective that improves any dual-encoder system.
Charith Chandra Sai Balne, Rithwik Maramraju, Siddharth Pratap Singh +4
Walmart Global Tech USA · Walmart Global Tech2026 USA
Dense retrieval systems increasingly face an asymmetry between complex instruction-like queries and relatively simple, static document collections. While stronger embedders can better understand such queries, re-embedding large corpora or fine-tuning large models is often impractical. We propose Efficient Retrieval Adapter (ERA), a query-side adapter learning framework for re-index-free retrieval adaptation. ERA first aligns the embedding spaces of a strong query embedder and a lightweight document embedder using unlabeled corpus documents, and then adapts the aligned query representation with a small number of labeled query-document pairs. Across 126 MAIR retrieval tasks from six domains, ERA improves average nDCG@10 by up to 8.2 points in symmetric settings and by more than 12 points in asymmetric settings, while using substantially fewer labels than supervised adapter training. These results show that retrieval systems can benefit from stronger query understanding without updating backbone embedders or rebuilding document indexes.
Dense retrievers excel at first-stage candidate generation but lack effective reranking in zero-resource settings. Existing approaches face a fundamental dilemma: cross-encoders deliver strong reranking quality but require costly supervised training and incur high latency, while unsupervised BM25 reranking consistently degrades dense retrieval performance on most of BEIR benchmarks. We propose DART (Dense Adaptive Reranking at Test-time), which resolves this dilemma by adapting the scoring function at inference time. For each query, the top-ranked documents serve as pseudo-positive examples and the bottom-ranked as pseudo-negative examples, providing noisy but readily available supervision to adapt a bilinear scoring matrix W via a small number of gradient updates. We further introduce a confidence-weighted margin loss and a cross-query momentum buffer that warm-starts adaptation across queries. On six BEIR benchmarks, DART achieves a mean per-dataset relative NDCG@10 gain of +2.1% over the dense retrieval baseline with under 10ms additional latency per query, demonstrating a powerful capability for zero-shot performance enhancement and cross-domain generalization.
Shiyan Liu, Yichen Li
Huazhong University of Science and Technology · ByteDance