固定秩与秩一偶阶对称张量分解的等价性
Equivalence of Fixed-Rank and Rank-One Even-Order Symmetric Tensor Factorization
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中文总结 AI 辅助
本文证明有限秩偶阶对称张量分解在贝叶斯最优设置下与秩一情形自由熵极限等价，通过信息论恒等式和复制对称性将多维变分公式约化为一维，并调整了基于特征值的方法以处理Hadamard幂。
中文摘要 AI 辅助
在Barbier、Ko及第二作者最近关于次线性秩对称矩阵分解的工作[Math. Stat. Learn. 9 (2026), 1-68]中，一个关键结果是：在贝叶斯最优设置下，当信号具有中心化独立同分布条目时，有限秩尖峰Wigner模型的自由熵的大尺寸极限与秩一情形相同。本文表明，这一秩一等价性结果可推广至有限秩、偶阶、对称张量分解的情形。此外，我们对前述工作中被声明为该结果必要条件的假设给出了一个自然的重新表述。与矩阵情形类似，我们使用信息论恒等式和复制对称性，将已知的极限自由熵多维变分公式约化为一维类比。新颖之处在于，该公式涉及的复制对称势包含变分参数（矩阵值）的Hadamard（逐元素）幂而非平方，因此必须调整矩阵情形中基于特征值的方法。
英文摘要
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.
发表机构
- The University of Turin（都灵大学）
- The University of Hong Kong（香港大学）
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