Z-Eigenvector Residual Recovery for Dominant Fourth-Order Structure Missed by PCA

Principal component analysis is a second-order method, it selects covariance dominant directions. In local data models, however, a structural component may be rare or intermittent and therefore have modest variance but large fourth-order response. This note demonstrates a straightforward fourth-order enhancement based on Z-eigenvectors of fourth-order tensors. We prove a separation result showing that, in a fourth-order-dominant regime, rank-k PCA selects nuisance directions, while successively selected fourth-order maximizing directions recover the structural subspace. In the noiseless case, the resulting structural projection has strictly smaller squared reconstruction error for every nonzero structural vector. For arbitrary deterministic errors, we give an explicit sufficient condition under which the same improvement holds. We also establish the stability of our residual recovery scheme under tensor perturbation. Numerical experiments include a controlled additive-noise study and an application-motivated synthetic baseball-swing example, illustrating the distinction between PCA and residual fourth-order recovery under perturbation and in mixed feature coordinates.