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

Ruba Hussen Morsi; Anas A. Rahman

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arXiv · 2609.06971

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

Ruba Hussen Morsi·Anas A. Rahman

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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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Ruba Hussen Morsi, Anas A. Rahman. 2026-09-07. Equivalence of Fixed-Rank and Rank-One Even-Order Symmetric Tensor Factorization. https://arxiv.org/abs/2609.06971

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