Comparison of scattering under reshaping

Establish a general comparison between the positive-scattering terms of a nonnegative tensor decomposition before and after grouping its modes through reshaping.

Background

Reshaping groups several tensor modes into composite modes and can increase the Lovitz–Petrov dimension budget. However, grouping also changes the factor-space geometry and therefore may alter the positive-scattering term in ways not controlled by dimension alone. The paper does not establish whether grouped and ungrouped scattering terms satisfy any general monotonicity or comparison relation.

References

We do not establish a general monotonicity relation between the scattering term before and after reshaping. In particular, the grouped scattering term may interact with the changed factor geometry in ways that are not captured by the dimension budget alone.

— Identifiability of Nonnegative Tensor Decompositions via Positive Scattering  (2609.11606 - Wang et al., 10 Sep 2026) in Section 6.1, Subsection “Reshaping”; also Conclusion