Equivalence of Fixed-Rank and Rank-One Even-Order Symmetric Tensor Factorization

Abstract

In the recent work of Barbier, Ko, and the second present author on sublinear-rank symmetric matrix factorization [Math. Stat. Learn. 9 (2026), 1-68], a key result is that, in the Bayes-optimal setting, the large-size limit of the free entropy of the finite-rank spiked Wigner model is the same as in the rank-one case when the signal has centered i.i.d. entries. In this paper, we show that this rank-one equivalence result extends to the case of finite-rank, even-order, symmetric tensor factorization. Moreover, we give a natural reformulation of a hypothesis that was stated in the aforementioned work to be necessary for this result. As in the matrix case, we use information-theoretic identities and replica symmetry to reduce a known multi-dimensional variational formula for the limiting free entropy to its one-dimensional analog. The novelty stems from the fact that said formula involves a replica symmetric potential containing Hadamard (entrywise) powers, rather than squares, of the matrix-valued variational parameter, so the eigenvalue-based approach used in the matrix case must be adjusted.

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BibTeX RIS

BibTeX

@misc{morsi2026equivalence,
  title = {{Equivalence of Fixed-Rank and Rank-One Even-Order Symmetric Tensor Factorization}},
  author = {Ruba Hussen Morsi and Anas A. Rahman},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.06971},
  eprint = {2609.06971},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.06971},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - Equivalence of Fixed-Rank and Rank-One Even-Order Symmetric Tensor Factorization
AU  - Ruba Hussen Morsi
AU  - Anas A. Rahman
PY  - 2026
DA  - 2026/09/07
PB  - arXiv
DO  - 10.48550/arxiv.2609.06971
UR  - https://arxiv.org/abs/2609.06971
AB  - In the recent work of Barbier, Ko, and the second present author on sublinear-rank symmetric matrix factorization [Math. Stat. Learn. 9 (2026), 1-68], a key result is that, in the Bayes-optimal setting, the large-size limit of the free entropy of the finite-rank spiked Wigner model is the same as in the rank-one case when the signal has centered i.i.d. entries. In this paper, we show that this rank-one equivalence result extends to the case of finite-rank, even-order, symmetric tensor factorization. Moreover, we give a natural reformulation of a hypothesis that was stated in the aforementioned work to be necessary for this result. As in the matrix case, we use information-theoretic identities and replica symmetry to reduce a known multi-dimensional variational formula for the limiting free entropy to its one-dimensional analog. The novelty stems from the fact that said formula involves a replica symmetric potential containing Hadamard (entrywise) powers, rather than squares, of the matrix-valued variational parameter, so the eigenvalue-based approach used in the matrix case must be adjusted.
ER  -

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