The Global Topology of Orthogonally Decomposable Tensor Landscapes
Abstract
The homogeneous form associated with a symmetric tensor, restricted to the sphere, is the objective of the best rank-one approximation problem and, with random coefficients, the energy of a mean-field spin glass. Its critical points have been studied extensively, but the global organization of the landscape they form is far less understood. We study this global structure for positive orthogonally decomposable tensors. We determine the persistent homology of the sublevel and superlevel filtrations in closed form, for every homological dimension q, every ambient dimension D and every tensor order k, via a recurrence in the ambient dimension. Taking the coefficients to be random, we further prove laws of large numbers for the resulting persistence diagrams at dimension $0$ and D−2, giving an exact description of the typical global topology of a spin-glass-like energy landscape. We illustrate our results with numerical computations and simulations.
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BibTeX
@misc{siu2026global,
title = {{The Global Topology of Orthogonally Decomposable Tensor Landscapes}},
author = {Chunyin Siu},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.14186},
eprint = {2609.14186},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.14186},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - The Global Topology of Orthogonally Decomposable Tensor Landscapes
AU - Chunyin Siu
PY - 2026
DA - 2026/09/12
PB - arXiv
DO - 10.48550/arxiv.2609.14186
UR - https://arxiv.org/abs/2609.14186
AB - The homogeneous form associated with a symmetric tensor, restricted to the sphere, is the objective of the best rank-one approximation problem and, with random coefficients, the energy of a mean-field spin glass. Its critical points have been studied extensively, but the global organization of the landscape they form is far less understood. We study this global structure for positive orthogonally decomposable tensors. We determine the persistent homology of the sublevel and superlevel filtrations in closed form, for every homological dimension q, every ambient dimension D and every tensor order k, via a recurrence in the ambient dimension. Taking the coefficients to be random, we further prove laws of large numbers for the resulting persistence diagrams at dimension $0$ and D−2, giving an exact description of the typical global topology of a spin-glass-like energy landscape. We illustrate our results with numerical computations and simulations.
ER -