基于预条件度量下张量列车低秩商几何的张量补全
Tensor Completion via Tensor Train Based Low-Rank Quotient Geometry under a Preconditioned Metric
AI总结：
本文将预条件度量从矩阵推广到张量列车格式，研究固定张量列车秩流形的商几何，并提出黎曼梯度下降、共轭梯度及高斯-牛顿等张量补全算法，实验验证其恢复能力、收敛性能与重建质量具有竞争力。
摘要：
本文研究低秩张量补全问题，即从部分观测条目中恢复张量。我们在张量列车（Tensor Train）格式下考虑该问题，并将预条件度量从矩阵情形推广到张量情形。详细研究了该度量下固定张量列车秩张量流形的一阶和二阶商几何。基于该商几何，提出了包括黎曼梯度下降、黎曼共轭梯度和黎曼高斯-牛顿在内的张量补全算法。还证明了商几何上的黎曼高斯-牛顿方法等价于具有特定收缩的嵌入几何上的黎曼高斯-牛顿方法。在随机实例以及与函数相关张量上的实验评估表明，所提算法在恢复能力、收敛性能和重建质量方面与现有其他算法具有竞争力。
英文摘要：
This paper investigates the low-rank tensor completion problem, which is about recovering a tensor from partially observed entries. We consider this problem in the tensor train format and extend the preconditioned metric from the matrix case to the tensor case. The first-order and second-order quotient geometry of the manifold of fixed tensor train rank tensors under this metric is studied in detail. Algorithms, including Riemannian gradient descent, Riemannian conjugate gradient, and Riemannian Gauss-Newton, have been proposed for the tensor completion problem based on the quotient geometry. It has also been shown that the Riemannian Gauss-Newton method on the quotient geometry is equivalent to the Riemannian Gauss-Newton method on the embedded geometry with a specific retraction. Empirical evaluations on random instances as well as on function-related tensors show that the proposed algorithms are competitive with other existing algorithms in terms of recovery ability, convergence performance, and reconstruction quality.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
