正交可分解张量景观的全局拓扑
The Global Topology of Orthogonally Decomposable Tensor Landscapes
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中文总结 AI 辅助
本研究针对正正交可分解张量景观，通过递推确定其子水平与超水平过滤的持续同调，并在随机系数下证明持久图的强大数定律，精确刻画了自旋玻璃类能量景观的典型全局拓扑。
中文摘要 AI 辅助
对称张量相关的齐次形式限制在球面上，是最佳秩一逼近问题的目标函数，在随机系数下，也是平均场自旋玻璃的能量。其临界点已被广泛研究，但它们所形成的景观的全局组织却远未被理解。我们研究了正正交可分解张量的这种全局结构。我们确定了子水平和超水平过滤的持续同调，以闭合形式给出，适用于每个同调维度 $q$、每个环境维度 $D$ 和每个张量阶 $k$，通过环境维度中的递推实现。在系数随机的情况下，我们进一步证明了在维度 $0$ 和 $D-2$ 处所得持久图的强大数定律，从而精确描述了自旋玻璃类能量景观的典型全局拓扑。我们通过数值计算和模拟说明了我们的结果。
英文摘要
The homogeneous form associated with a symmetric tensor, restricted to the sphere, is the objective of the best rank-one approximation problem and, with random coefficients, the energy of a mean-field spin glass. Its critical points have been studied extensively, but the global organization of the landscape they form is far less understood. We study this global structure for positive orthogonally decomposable tensors. We determine the persistent homology of the sublevel and superlevel filtrations in closed form, for every homological dimension $q$, every ambient dimension $D$ and every tensor order $k$, via a recurrence in the ambient dimension. Taking the coefficients to be random, we further prove laws of large numbers for the resulting persistence diagrams at dimension $0$ and $D-2$, giving an exact description of the typical global topology of a spin-glass-like energy landscape. We illustrate our results with numerical computations and simulations.
