AI 中文总结
研究正交群上按元素的\(\ell_4\)范数的局部极大值，通过基于正交随机矩阵\(Q^{\circ2}\)的最大元素枢轴法，证明带符号置换矩阵是唯一局部及全局极大值点，其他驻点有特定切向方向，论证自成体系处理多种情况。
AI 中文摘要
我们对实正交群\(\mathcal O(r)\)上按元素的四次方目标函数\(Q\longmapsto \lVert Q\rVert_4^4 =\sum_{i,j=1}^r q_{ij}^4\)的局部极大值进行分类。证明了在每个维度中，带符号置换矩阵是唯一的局部极大值点，因此也是唯一的全局极大值点。更强的是，其他每个驻点都有一个明确的秩二切向方向且二阶变分严格为正。证明基于正交随机矩阵\(Q^{\circ2}\)的最大元素枢轴法，相关的全黎曼黑塞矩阵可精确求值且在最大非单位平方元素处为正。该论证自成体系，处理了零、重复大小、可约支撑和哈达玛型驻点等情况。
英文摘要
We classify the local maximizers of the entrywise fourth-power objective \[ Q\longmapsto \lVert Q\rVert_4^4 =\sum_{i,j=1}^r q_{ij}^4 \] over the real orthogonal group $\mathcal O(r)$. We prove that the signed permutation matrices are the only local maximizers, and hence the only global maximizers, in every dimension. More strongly, every other stationary point has an explicit rank-two tangent direction with strictly positive second variation. The proof is based on a maximum-entry pivot for the orthostochastic matrix $Q^{\circ2}$: the associated full Riemannian Hessian can be evaluated exactly and is positive at a largest nonunit squared entry. The argument is self-contained and handles zeros, repeated magnitudes, reducible support, and Hadamard-type stationary points.
