🤖 AI Summary
This work addresses the limitation of conventional element-wise reconstruction error in tensor low-rank approximation, which fails to capture geometric degradation of multidimensional structures. Building upon the orthogonal Tucker model, the paper introduces a novel orthogonal decomposition of reconstruction error into directional loss—quantifying subspace deviations caused by truncation and noise—and interaction loss—measuring distortions in the multilinear interactions of the core tensor. A Wedin-type stability bound is established for the directional loss. Experiments on synthetic and hyperspectral data, leveraging matrix SVD, tensor Tucker decomposition, and subspace perturbation analysis, demonstrate that under identical reconstruction errors, directional loss can vary by up to 4.6×, and high directional loss strongly correlates with visual blurring, thereby underscoring the necessity of structure-aware error metrics.
📝 Abstract
Matricized low-rank approximation via SVD is a standard surrogate for tensor decompositions, but entry-wise reconstruction error fails to capture multiway geometric degradation. Under an orthogonal Tucker model, we characterize this degradation using two metrics: cross-mode Direction Loss, measuring geometric subspace deviation from rank truncation and noise rotation, and Interaction Loss, quantifying multilinear interaction distortion in the core tensor. We prove that squared relative reconstruction error orthogonally decomposes into interaction loss and out-of-subspace energy, and derive a Wedin-type bound establishing the stability of a plug-in Direction Loss estimator. Experiments on synthetic and hyperspectral datasets demonstrate that nearly identical reconstruction errors can yield markedly different structural-loss profiles; hyperspectral patches with comparable reconstruction errors exhibit up to a 4.6-fold difference in Direction Loss, correlating with severe visual blurring.