正则单纯形张量：优化景观、猜想证明及其延伸

中文摘要 AI 辅助

张量特征对的概念在过去几十年中吸引了越来越多的研究关注。近期工作聚焦于一类特殊的张量，称为正则单纯形张量，它由n维空间中n个向量的等角紧框架构造，其中n≥3且阶数m≥3。现有工作主要分析通过张量幂法获得的特征对的鲁棒性。在那项工作的结尾，提出了一个猜想：如果存在，正则单纯形张量的唯一鲁棒特征向量（在符号等价意义下）是正则单纯形框架中的向量。后续研究从理论上证明了这一猜想在n=3的最简单三角形情形下成立。然而，对于更高的n，在检查所有特征对以及确定鲁棒性准则的显式公式方面，过程变得复杂。在本文中，为解决这一问题，我们建立了鲁棒特征对与局部最优特征对之间的联系，后者被认为是优化领域中的另一个关键概念。然后，我们转向检查所有特征对的局部最优性，为此我们开发了一个具有良好结构的有效模型，该模型便于枚举所有特征对并描绘该模型的优化景观。接着，整合这两项进展使我们能够将鲁棒特征对的范围缩小到那些局部最大化的特征对，这些特征对恰好是正则单纯形框架中的向量。最后，猜想的证明简化为仅检查框架中的向量，其鲁棒性对于任何更高的n和m都可以轻松检查。这项工作表明，在符号等价意义下，除了不存在鲁棒特征对的例外情形(m,n)=(3,3)/(3,4)/(4,3)之外，正则单纯形张量的唯一鲁棒特征向量是正则单纯形框架中的向量。

英文摘要

The concept of tensor eigenpairs has attracted increasing research attention in the past decades. Recent works have focused on a special class termed regular simplex tensors, which are constructed from an equiangular tight frame of n vectors in (n-1) dimensional space for n >= 3 and order m >= 3. Existing works focus on analyzing the robustness of eigenpairs obtained by the tensor power method. At the end of that work, a conjecture was made that if they exist, the only robust eigenvectors of a regular simplex tensor, up to sign equivalence, are the vectors in the regular simplex frame. A subsequent study theoretically proved that this holds for the simplest triangle case where n = 3. However, for cases with higher n, the process becomes complicated in both checking all eigenpairs and determining the explicit formula for the robustness criterion. In this paper, to deal with this issue, a connection between robust and locally optimal eigenpairs is built, recognizing the latter as another pivotal concept in the field of optimization. Then, we turn to checking the local optimality of all eigenpairs, for which we have developed an efficient model with a favorable structure that facilitates the enumeration of all eigenpairs and delineates the optimization landscape for the model. Then, integrating the two advances enables us to narrow the scope of the robust eigenpairs to those locally maximized ones, which are exactly the vectors in the regular simplex frame. Finally, the proof of the conjecture reduces to only examining the vectors in the frame, whose robustness can be easily checked for any higher n and m. This work shows that, up to sign equivalence, excluding the exceptional cases (m, n) = (3, 3)/(3, 4)/(4, 3) where no robust eigenpairs exist, the only robust eigenvectors of a regular simplex tensor are the vectors in the regular simplex frame.

发表机构

• the School of Microelectronics and Communication Engineering, Chongqing University（重庆大学微电子与通信工程学院）

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