In this paper, we provide the first convergence guarantee for the
factorization approach. Specifically, to avoid the scaling ambiguity and to
facilitate theoretical analysis, we optimize over the so-called left-orthogonal
TT format which enforces orthonormality among most of the factors. To ensure
the orthonormal structure, we utilize the Riemannian gradient descent (RGD) for
optimizing those factors over the Stiefel manifold. We first delve into the TT
factorization problem and establish the local linear convergence of RGD.
Notably, the rate of convergence only experiences a linear decline as the
tensor order increases. We then study the sensing problem that aims to recover
a TT format tensor from linear measurements. Assuming the sensing operator
satisfies the restricted isometry property (RIP), we show that with a proper
initialization, which could be obtained through spectral initialization, RGD
also converges to the ground-truth tensor at a linear rate. Furthermore, we
expand our analysis to encompass scenarios involving Gaussian noise in the
measurements. We prove that RGD can reliably recover the ground truth at a
linear rate, with the recovery error exhibiting only polynomial growth in
relation to the tensor order. We conduct various experiments to validate our
theoretical findings.

通过对所谓的左正交 TT 格式进行 Riemannian 梯度下降（RGD）优化，本论文首次为因子分解方法提供了收敛保证，并证明了 sensing 问题中 RGD 在适当初始化下能够以线性速率收敛到真值张量。

张量列车恢复的保证非凸分解方法

Guaranteed Nonconvex Factorization Approach for Tensor Train Recovery

We introduce a novel random projection technique for efficiently reducing the
dimension of very high-dimensional tensors. Building upon classical results on
Gaussian random projections and Johnson-Lindenstrauss transforms~(JLT), we
propose two tensorized random projection maps relying on the tensor train~(TT)
and CP decomposition format, respectively. The two maps offer very low memory
requirements and can be applied efficiently when the inputs are low rank
tensors given in the CP or TT format. Our theoretical analysis shows that the
dense Gaussian matrix in JLT can be replaced by a low-rank tensor implicitly
represented in compressed form with random factors, while still approximately
preserving the Euclidean distance of the projected inputs. In addition, our
results reveal that the TT format is substantially superior to CP in terms of
the size of the random projection needed to achieve the same distortion ratio.
Experiments on synthetic data validate our theoretical analysis and demonstrate
the superiority of the TT decomposition.

介绍了一种基于张量分解格式 —— 张量列车（TT）和 CP 分解格式 —— 的两种张量随机投影映射方法来降低高维张量的维数，从而显著地降低了存储和计算成本，并证明了采用 TT 格式比 CP 格式更能够更快地实现相同的降维效果，相关实验验证了理论分析结果。

张量随机投影

Tensorized Random Projections

The hierarchical SVD provides a quasi-best low rank approximation of high
dimensional data in the hierarchical Tucker framework. Similar to the SVD for
matrices, it provides a fundamental but expensive tool for tensor computations.
In the present work we examine generalizations of randomized matrix
decomposition methods to higher order tensors in the framework of the
hierarchical tensors representation. In particular we present and analyze a
randomized algorithm for the calculation of the hierarchical SVD (HSVD) for the
tensor train (TT) format.

该研究针对层次张量表示，研究了随机矩阵分解方法在高阶张量中的推广，并提出分析了一种用于计算张量拓扑结构的随机算法。