Spatial-sign-based multilinear principal component analysis for tensor data

2026-08-30

AI Summary

为解决MPCA在重尾分布和污染下的不稳定性，提出基于空间符号的多线性主成分分析(SMPCA)，通过空间中位数中心化、空间符号归一化及交替特征分解来稳健降维。

Abstract

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.

Problem

• multilinear principal component analysis
• heavy-tailed distributions
• contamination

Innovation

• spatial-sign-based
• multilinear principal component analysis
• robust dimension reduction
• tensor data
• heavy-tailed distributions