Equivalence found in complex tensor and matrix factorization models
Equivalence of Fixed-Rank and Rank-One Even-Order Symmetric Tensor Factorization
Information Theory
Summary
Understanding how complicated data can be broken down into simpler parts is important in many fields like machine learning. This paper shows that a special way to break down data arranged in even-order symmetric tensors behaves the same as when the data is just rank-one, which is simpler. Building on recent work with matrices, the authors extend the idea to more complex data structures using advanced math tools. Their work also clarifies an earlier assumption needed for this equivalence to hold, adjusting techniques to fit the new setting.
symmetric tensor factorizationrank-one equivalenceBayes-optimal settingfree entropyspiked Wigner modelreplica symmetryHadamard powersvariational formula
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.