Rank equivalence found in multi-dimensional symmetric tensor factorization
Equivalence of Fixed-Rank and Rank-One Even-Order Symmetric Tensor Factorization
Information Theory
Summary
Factoring complex data structures called tensors can be hard, especially when they have many dimensions. The authors show that breaking down even-order symmetric tensors of finite rank is fundamentally the same as breaking down rank-one tensors in certain statistical settings. They extend previous math results from matrices to these more complex shapes. This finding simplifies understanding of tensor factorization by reducing complex multi-dimensional problems to simpler one-dimensional ones.
What this means in practice
- •For signal processing engineers: Design algorithms for extracting signals from multi-dimensional symmetric data using rank-one equivalent simplifications.
- •For machine learning practitioners: Improve tensor-based models by applying reduced-rank factorizations equivalent to simpler rank-one cases for inference tasks.
A theory result. No direct application yet.
Authors
Ruba Hussen Morsi, Anas A. Rahman
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.