Structural Loss Metrics for Tensor Approximation via Matrix Low-Rank Approximation
2026-07-27 • Computer Vision and Pattern Recognition
Computer Vision and Pattern Recognition
AI summaryⓘ
The authors study how standard low-rank approximations of tensors, usually judged by simple entry-wise errors, miss important geometric details in the data. They introduce two new ways to measure this loss: one checks how the directions in the data shift (Direction Loss), and the other looks at how interactions within the data structure change (Interaction Loss). They also prove mathematical relationships between these losses and traditional error measures, and show their method works well on simulated and real hyperspectral images. Their results reveal that similar usual errors can hide very different impacts on data geometry, which can explain visible differences like blurring.
tensor decompositionlow-rank approximationsingular value decomposition (SVD)Tucker modelmultilinear algebrareconstruction errorDirection LossInteraction Losshyperspectral imagingWedin bound
Authors
Hiroki Hasegawa
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.