Tensor decompositions such as the canonical format and the tensor train
format have been widely utilized to reduce storage costs and operational
complexities for high-dimensional data, achieving linear scaling with the input
dimension instead of exponential scaling. In this paper, we investigate even
lower storage-cost representations in the tensor ring format, which is an
extension of the tensor train format with variable end-ranks. Firstly, we
introduce two algorithms for converting a tensor in full format to tensor ring
format with low storage cost. Secondly, we detail a rounding operation for
tensor rings and show how this requires new definitions of common linear
algebra operations in the format to obtain storage-cost savings. Lastly, we
introduce algorithms for transforming the graph structure of graph-based tensor
formats, with orders of magnitude lower complexity than existing literature.
The efficiency of all algorithms is demonstrated on a number of numerical
examples, and in certain cases, we demonstrate significantly higher compression
ratios when compared to previous approaches to using the tensor ring format.

本文探讨如何使用张量分解技术，通过引入张量环格式、设计新算法、定义线性代数操作并实现改变张量结构等方式，获得更低的存储成本和更高的压缩比。

张量环分解及其计算算法

On Algorithms for and Computing with the Tensor Ring Decomposition

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。