研究多模态数据缺失对PLS分析的影响，发现存在信号恢复的临界阈值。
Missing-Data-Induced Phase Transitions in Spectral PLS for Multimodal Learning
- 在高维稀疏模型下分析带缺失数据的PLS-SVD，揭示其行为类似衰减信号的随机矩阵。
- 当信号强度超过临界值时，主奇异向量能有效捕捉潜在共享结构，低于则无效。
- 适用于多模态学习中数据缺失场景的理论分析，适合关注统计推断的科研人员。
偏最小二乘法（PLS）通过经验交叉协方差的前导奇异向量学习多模态数据的共享结构（PLS-SVD），但实际数据常存在双视图缺失。本文在比例高维尖峰模型下，研究独立逐项缺失（完全随机缺失）情况下的PLS-SVD。经适当归一化后，遮蔽交叉协方差表现为有效信号强度被√ρ衰减的尖峰矩形随机矩阵，其中ρ为联合条目保留概率。复制对称分析预测了一个类似BBP的相变：低于临界信噪比时，主奇异向量渐近无信息；高于该阈值时，与潜在共享方向实现非平凡对齐，并给出闭式渐近重叠公式。还提出有限秩扩展猜想，认为当潜变量分离时，同一调整后的缺失阈值在各分量上均适用。模拟与半合成多模态实验验证了预测的相图和恢复曲线，涵盖不同纵横比、信号强度及缺失水平。
原文摘要 · Abstract (English)
Partial Least Squares (PLS) learns shared structure from paired data via the top singular vectors of the empirical cross-covariance (PLS-SVD), but multimodal datasets often have missing entries in both views. We study PLS-SVD under independent entry-wise missing-completely-at-random masking in a proportional high-dimensional spiked model. After appropriate normalization, the masked cross-covariance behaves like a spiked rectangular random matrix whose effective signal strength is attenuated by $\sqrtρ$, where $ρ$ is the joint entry retention probability. The replica-symmetric analysis predicts a sharp BBP-type phase transition: below a critical signal-to-noise threshold the leading singular vectors are asymptotically uninformative, while above it they achieve nontrivial alignment with the latent shared directions, with closed-form asymptotic overlap formulas. We also state a finite-rank extension as a conjecture, predicting that the same missingness-adjusted threshold applies componentwise when the latent spikes are separated. Simulations and semi-synthetic multimodal experiments agree with the predicted phase diagram and recovery curves across aspect ratios, signal strengths, and missingness levels.
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