In this paper we propose an efficient method to compress a high dimensional
function into a tensor ring format, based on alternating least-squares (ALS).
Since the function has size exponential in $d$ where $d$ is the number of
dimensions, we propose efficient sampling scheme to obtain $O(d)$ important
samples in order to learn the tensor ring. Furthermore, we devise an
initialization method for ALS that allows fast convergence in practice.
Numerical examples show that to approximate a function with similar accuracy,
the tensor ring format provided by the proposed method has less parameters than
tensor-train format and also better respects the structure of the original
function.

本文提出了一种基于交替最小二乘法的高效方法来将高维函数压缩成张量环格式，同时采用有效的采样方案获得 $O (d)$ 个重要样本来学习张量环，并提出了 ALS 的初始化方法以便快速收敛。 数值实验表明，相比于张量列车格式，所提出的张量环格式具有更少的参数，并且更好地保持了原始函数的结构。

从采样中高效构建张量环表示

Efficient construction of tensor ring representations from sampling

For reconstruction of low-rank matrices from undersampled measurements, we
develop an iterative algorithm based on least-squares estimation. While the
algorithm can be used for any low-rank matrix, it is also capable of exploiting
a-priori knowledge of matrix structure. In particular, we consider linearly
structured matrices, such as Hankel and Toeplitz, as well as positive
semidefinite matrices. The performance of the algorithm, referred to as
alternating least-squares (ALS), is evaluated by simulations and compared to
the Cram\'er-Rao bounds.

基于最小二乘估计的迭代算法可以用于重建低秩矩阵，并且针对线性结构的矩阵和正半定矩阵等具有先验知识的矩阵，有更好的性能，称为交替最小二乘 (ALS) 算法，并通过模拟实验和 Cramér-Rao 下界进行了比较。