发表机构
Nankai University; Dongbei University of Finance and Economics; Hong Kong Baptist University（南开大学; 东北财经大学; 香港浸会大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
针对MPCA在重尾分布和污染下不稳定的问题，提出SMPCA方法，通过空间符号归一化等实现稳健降维，经理论和实验验证其在重尾场景下的准确性与稳定性优于现有方法。
AI 中文摘要
多重主成分分析（MPCA）可降低张量值数据的维度，同时保留其模式特异性结构，但其二次散度准则在重尾分布和污染情况下可能不稳定。我们提出基于空间符号的多重主成分分析（SMPCA），这是一种稳健的降维方法，通过空间中位数对观测值进行中心化，经空间符号归一化去除径向幅度，并通过交替特征分解估计各模式的载荷空间。在可分张量椭圆模型下，我们证明目标模式载荷空间可唯一最大化总体准则，且一次完整的精确总体块更新可从任意初始化中恢复这些空间。我们还精确刻画了它们的张量积子空间何时与向量化空间符号PCA的主导无约束子空间重合，以及当存在有限二阶矩时，何时与普通向量化PCA重合。在样本层面，我们推导了模式子空间和联合多重线性投影器的显式统计速率，获得了相应的重构保证，建立了累积贡献维度选择器的一致性，并证明精确循环更新生成的目标值是非递减且收敛的。模拟和实证应用表明，SMPCA在重尾分布和异常值污染下比竞争对手更准确、更稳定，同时在轻尾设置下保持有竞争力的性能。
英文摘要
Multilinear principal component analysis (MPCA) reduces the dimension of tensor-valued data while preserving their mode-specific structure, but its quadratic scatter criterion can be unstable under heavy-tailed distributions and contamination. We propose spatial-sign-based multilinear principal component analysis (SMPCA), a robust dimension-reduction method that centers the observations by their spatial median, removes radial magnitude through spatial-sign normalization, and estimates the mode-wise loading spaces by alternating eigendecompositions. Under a separable tensor elliptical model, we show that the target mode-wise loading spaces uniquely maximize the population criterion and that one complete sweep of exact population block updates recovers them from any initialization. We also characterize exactly when their tensor-product subspace coincides with a leading unrestricted subspace of vectorized spatial-sign PCA and, when finite second moments exist, ordinary vectorized PCA. At the sample level, we derive explicit statistical rates for the mode-wise subspaces and the joint multilinear projector, obtain corresponding reconstruction guarantees, establish consistency of the cumulative-contribution dimension selector, and prove that the objective values generated by exact cyclic updates are nondecreasing and convergent. Simulations and an empirical application show that SMPCA is more accurate and stable than competitors under heavy-tailed distributions and outlier contamination, while retaining competitive performance under light-tailed settings.
