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.

翻译：本文首次提供了因子化方法的收敛性保证。具体而言，为避免尺度模糊性并便于理论分析，我们优化了所谓的左正交TT格式，该格式强制大多数因子满足正交归一性。为确保正交归一结构，我们利用黎曼梯度下降法（RGD）在施蒂费尔流形上优化这些因子。我们首先深入研究了TT因子化问题，并建立了RGD的局部线性收敛性。值得注意的是，收敛速度仅随张量阶数的增加而线性下降。随后，我们研究了旨在从线性测量中恢复TT格式张量的感知问题。假设感知算子满足受限等距性质（RIP），我们证明通过谱初始化获得适当初始值后，RGD也能以线性速率收敛到真实张量。此外，我们将分析扩展到测量中包含高斯噪声的场景。我们证明RGD能够以线性速率可靠地恢复真实值，且恢复误差相对于张量阶数仅呈多项式增长。我们进行了多种实验来验证理论发现。