发表机构
School of Mathematics and Statistics, Southwest University（西南大学数学与统计学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出一种自适应块加权模向黎曼梯度下降方法，用于从线性测量中恢复低多线性秩张量，通过归一化加权策略改善收敛性，并在理论上建立线性收敛和采样保证，实验验证其高效性和可靠性。
AI 中文摘要
我们考虑从线性测量中恢复低多线性秩张量的问题，并提出一种自适应块加权模向黎曼梯度下降方法。该方法将内存高效的模向测量与针对黎曼梯度的核心和因子分量的归一化自适应加权策略相结合。该加权策略在不增加搜索方向的多线性秩界限或用于回缩的缩减核心大小的情况下改善了收敛性。在张量受限等距性质和合适的初始化条件下，我们建立了局部线性收敛性，并为亚高斯和带随机符号的子采样正交（SORS）测量导出了采样保证。在合成低Tucker秩张量上的数值实验表明，所提出的方法减少了迭代次数和计算时间，同时保持了可靠的恢复性能，尤其是在恢复阈值附近和结构化SORS测量情况下。
英文摘要
We consider the recovery of low-multilinear-rank tensors from linear measurements and propose an adaptive block-weighted modewise Riemannian gradient descent method. The method combines memory-efficient modewise measurements with a normalized adaptive weighting strategy for the core and factor components of the Riemannian gradient. The weighting improves convergence without increasing the multilinear-rank bound of the search direction or the size of the reduced core used for retraction. Under the tensor restricted isometry property and a suitable initialization, we establish local linear convergence and derive sampling guarantees for sub-Gaussian and subsampled orthogonal with random sign (SORS) measurements. Numerical experiments on synthetic low-Tucker-rank tensors show that the proposed method reduces iteration counts and computational time while maintaining reliable recovery performance, especially near the recovery threshold and for structured SORS measurements.
