稠密检索何时需要非对称几何？共享与双投影的偏差-方差理论
When Does Dense Retrieval Need Asymmetric Geometry? A Bias-Variance Theory of Shared and Dual Projections
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中文总结 AI 辅助
本文提出偏差-方差理论，阐明稠密检索中共享与双投影的适用条件，并设计CARS选择器，实验证明其能根据数据规模自适应选择最优几何，显著降低遗憾。
中文摘要 AI 辅助
稠密检索支撑着检索增强生成、语义搜索和问答系统，然而，在共享与双查询-文档投影之间进行选择的理论基础仍不明确。我们提出了低秩双线性评分的偏差-方差理论。共享投影产生正半定算子，而双投影实现任意低秩算子。我们推导了它们的精确逼近差距，并证明了一个局部高斯边界：当平方方向信号超过其额外自由度的估计成本时，双投影的风险恰好更低。该边界催生了交叉拟合非对称风险选择器（CARS），它从训练对中估计可复现的方向信号；其高斯对应形式具有精确的选择功效和遗憾公式。在理论指导下，我们在多个数据集和嵌入模型上进行了检索实验。随着查询旋转从0度增加到90度，双投影减去共享投影的平均NDCG@10优势增加了一倍以上。在秩-样本量网格中，在n=32时，共享投影在16个单元中赢得13个，而在n=1024和n=2048时，双投影赢得全部32个单元。与这种转变一致，所有168条可比较的算子风险曲线都随着训练数据的增长而向双投影移动。与两个固定几何基线相比，CARS将留出遗憾减少了49-96%，并实现了90.1%的平均几何选择准确率。
英文摘要
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.
发表机构
- The Hong Kong Polytechnic University（香港理工大学）
机构由 AI 辅助整理，请以论文原文为准。
