揭示了部分对齐信号下PLS方法的理论极限与失效条件。
Spectral Thresholds in Correlated Spiked Models and Fundamental Limits of Partial Least Squares
- 通过随机矩阵理论分析跨协方差模型中的谱阈值现象。
- 发现当信噪比低于临界值时，PLS完全无法恢复信号。
- 为高维多模态学习提供了可信赖方法的设计依据。
我们对具有部分对齐信号的高维双通道数据模型进行了严格的随机矩阵理论分析。这类模型源于多模态学习，是偏最小二乘法（PLS）的标准生成设定，而该方法虽广泛应用却缺乏理论支持。研究发现，样本交叉协方差矩阵的前导奇异值经历类似Baik-Ben Arous-Peche（BBP）的相变，可精确刻画信息成分出现的临界阈值。结果首次给出了此设定下PLS信号恢复能力的尖锐渐近描述，揭示了PLS与贝叶斯最优估计器之间的根本性能差距。特别地，我们识别出在某些信噪比（SNR）和相关性范围内，尽管信号理论上可检测，但PLS仍无法恢复任何信息。这些发现澄清了PLS的理论边界，并为高维多模态推断方法的设计提供指导。
原文摘要 · Abstract (English)
We provide a rigorous random matrix theory analysis of spiked cross-covariance models where the signals across two high-dimensional data channels are partially aligned. These models are motivated by multi-modal learning and form the standard generative setting underlying Partial Least Squares (PLS), a widely used yet theoretically underdeveloped method. We show that the leading singular values of the sample cross-covariance matrix undergo a Baik-Ben Arous-Peche (BBP)-type phase transition, and we characterize the precise thresholds for the emergence of informative components. Our results yield the first sharp asymptotic description of the signal recovery capabilities of PLS in this setting, revealing a fundamental performance gap between PLS and the Bayes-optimal estimator. In particular, we identify the SNR and correlation regimes where PLS fails to recover any signal, despite detectability being possible in principle. These findings clarify the theoretical limits of PLS and provide guidance for the design of reliable multi-modal inference methods in high dimensions.
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