Leakage and Multiway Interaction Loss: What Matricization and Single-Mode Truncation Discard from Multiway Signals

arXiv:2607.24009v2 Announce Type: replace Abstract: Tensor methods are said to preserve structure that matrix methods destroy, but what is lost, and why, is unclear. Relative to a multilinear reference, we split the squared error of any reconstruction exactly into coefficient distortion, multiway interaction loss, directional loss, and out-of-model leakage. Di rectional loss removes components in several modes; interaction loss removes a direction in one mode while its partners in other modes remain. Optimal truncation of a matrix or of an orthogonally decomposable tensor incurs only directional loss, whereas for generic tensors of order three or more with balanced ranks, optimally truncating one unfolding below its rank incurs only interaction loss. Matricization is the extreme case: when the other modes have low rank, it must under resolve the processed mode at budgets at which multilinear truncation is lossless, and under noise it adds leakage scaling with the product of the unprocessed dimensions. For sources with orthonormal signatures in all modes but one, interaction loss is the energy-weighted signature distortion of the surviving sources; without noise, a distortion below a threshold keeps the periodogram estimate within its main lobe and bounds its bias. To first order, multidimensional rotational-invariance estimation sees only in-model noise and single-mode leakage, so the leakage energy removed by tensor-based subspace estimation does not predict its gain. A calibrated resolution makes the split usable under noise. On a measured hyperspectral image with nearly full-rank spatial modes, spectral principal component analysis was as storage-efficient as multilinear truncation.