AI 中文总结
针对仅观测有限横向切片的结构化张量近似问题，提出BMTA算法，结合两种信号模型与低秩Tucker框架，通过理论分析与多数据集实验验证其有效性。
AI 中文摘要
在本研究中，我们考虑结构化张量近似问题，其中仅观测到有限数量的横向切片。所提出的算法称为基与流形先验张量近似（Basis and Manifold prior Tensor Approximation，BMTA），利用全局张量演化的全局与局部结构。具体而言，BMTA整合两种信号模型：（i）准基模型，用于捕获沿物理轨迹的平滑全局变化；（ii）流形引导插值模型，用于表征张量切片间的局部关系。结合低秩Tucker重构框架以高效捕获先验，得到基函数估计系数与张量优化。此外，我们提供理论分析，建立非渐近重构误差界，该界表征采样复杂度、优化收敛性及模型失配的影响。在合成与真实世界数据集上开展数值实验，包括量子化学及时空传感应用。
英文摘要
In this work, we consider a structured tensor approximation problem, where only a limited number of lateral slices are observed. The proposed algorithm , called Basis and Manifold prior Tensor Approximation (BMTA), exploits both global and local structures of the evolution of a global tensor. Specifically, BMTA integrates two signal models: (i) a quasi-basis model that captures smooth global variations along a physical trajectory, and (ii) a manifold-guided interpolation model that characterizes local relationships among tensor slices. A low-rank Tucker reconstruction framework is incorporated to efficiently capture the priors, resulting in coefficients for basis function estimation and a tensor optimization. In addition, we provide a theoretical analysis which establishes a non-asymptotic reconstruction error bound that characterizes the effects of sampling complexity, optimization convergence, and model mismatch. Numerical experiments are performed on both synthetic and real-world datasets, including quantum chemistry and spatiotemporal sensing applications.
