提出快速收敛的张量列车补全算法，大幅降低计算时间。
Fast and Provable Tensor-Train Format Tensor Completion via Precondtioned Riemannian Gradient Descent
- 用预条件黎曼梯度下降法优化张量列车格式补全
- 模拟数据上计算时间减少两个数量级
- 适合高维数据补全，如遥感图像与量子态重建
低秩张量补全旨在从部分观测数据中恢复完整张量，广泛应用于量子计算和图像处理等领域。由于张量列车（TT）格式在处理高阶结构化张量方面具有显著优势，本文研究基于TT格式的低秩张量补全问题。提出一种预条件黎曼梯度下降算法（PRGD），并证明其线性收敛性。在模拟与真实数据集上的实验表明，该算法效果显著：在模拟数据上，计算时间相比经典算法减少两个数量级；在高光谱图像补全和量子态层析等实际应用中，迭代次数大幅减少，从而显著降低计算耗时。
原文摘要 · Abstract (English)
Low-rank tensor completion aims to recover a tensor from partially observed entries, and it is widely applicable in fields such as quantum computing and image processing. Due to the significant advantages of the tensor train (TT) format in handling structured high-order tensors, this paper investigates the low-rank tensor completion problem based on the TT-format. We proposed a preconditioned Riemannian gradient descent algorithm (PRGD) to solve low TT-rank tensor completion and establish its linear convergence. Experimental results on both simulated and real datasets demonstrate the effectiveness of the PRGD algorithm. On the simulated dataset, the PRGD algorithm reduced the computation time by two orders of magnitude compared to existing classical algorithms. In practical applications such as hyperspectral image completion and quantum state tomography, the PRGD algorithm significantly reduced the number of iterations, thereby substantially reducing the computational time.
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