AI 中文总结
该研究揭示流行度偏差BPR训练下协同过滤用户嵌入会向噪声基底坍缩，推导了可计算的相变边界，实验表明部署级正则化下坍缩效应很小，且相关干预未提升推荐质量。
AI 中文摘要
我们研究了存在流行度偏差的BPR（贝叶斯个性化排序）训练如何重塑协同过滤嵌入的用户间几何结构。我们采用均值中心化的用户协方差$C=\ frac1n U^\ op H U$作为研究对象，该指标用于衡量用户之间的可区分程度，区别于以往研究使用的未中心化二阶矩。我们证明，在物品分布平稳的流行度偏差反馈下，$C$会收敛到与物品噪声协方差$Q$成比例的稳态，因此用户间离散度会向噪声基底坍缩。我们推导了训练超参数$(α,λ_{neg},γ,d)$下区分收缩与扩张的闭式可计算相变边界，并在MovieLens-25M数据集上验证了两个方向的预测。随后我们探究了该效应的局限：在部署级正则化强度下，预测的收缩效应真实存在且由策略驱动，但幅度较小，且未在我们测量的任何推荐层面指标中体现。由$α$驱动的各向异性坍缩机制仅在会降低推荐系统性能的正则化强度下才会生效。基于该理论提出的部署时恢复干预措施并未提升推荐质量。该边界可通过训练后模型的嵌入、物品交互计数和训练超参数计算得出，因此从业者无需模拟反馈循环即可检查已部署系统是否处于强坍缩区间。在我们的实验中，可部署的参数设置远未达到该区间。
英文摘要
We study how popularity-biased BPR training reshapes the between-user geometry of collaborative-filtering embeddings. We work with the mean-centered user covariance $C=\tfrac1n U^\top H U$, the object that measures how distinguishable users are from one another, as opposed to the uncentered second moment used in prior work. We prove that under popularity-biased feedback with stationary items, $C$ converges to a steady state proportional to the item-noise covariance $Q$. Thus between-user spread collapses toward a noise floor. We derive a closed-form, computable phase boundary in the training hyperparameters $(α,λ_{neg},γ,d)$ separating contraction from expansion, and validate both directional predictions on MovieLens-25M. We then examine the limits of the effect. At deployment-scale regularization the predicted contraction is real and policy-driven but small, and it is not reflected in any recommendation-level metric we measured. The $α$-driven anisotropic-collapse mechanism operates only at regularization strengths that degrade the recommender. A deployment-time restoration intervention derived from the theory does not improve recommendation quality. The boundary is computable from a trained model's embeddings, item interaction counts, and training hyperparameters, so a practitioner can check whether a deployed system sits in the strong-collapse regime without simulating the feedback loop. In our experiments the boundary places deployable settings far from that regime.
Comments7 pages, 2 figures
