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 pattern classification, polynomial classifiers are well-studied methods as
they are capable of generating complex decision surfaces. Unfortunately, the
use of multivariate polynomials is limited to kernels as in support vector
machines, because polynomials quickly become impractical for high-dimensional
problems. In this paper, we effectively overcome the curse of dimensionality by
employing the tensor train format to represent a polynomial classifier. Based
on the structure of tensor trains, two learning algorithms are proposed which
involve solving different optimization problems of low computational
complexity. Furthermore, we show how both regularization to prevent overfitting
and parallelization, which enables the use of large training sets, are
incorporated into these methods. Both the efficiency and efficacy of our
tensor-based polynomial classifier are then demonstrated on the two popular
datasets USPS and MNIST.

该研究利用张量列车格式来表示多项式分类器，提出了两种学习算法来解决低计算复杂度的优化问题，进而演示了其在实际应用数据集上的高效性和功效。

多项式分类器的并行张量列车学习

Parallelized Tensor Train Learning of Polynomial Classifiers

Deep neural networks currently demonstrate state-of-the-art performance in
several domains. At the same time, models of this class are very demanding in
terms of computational resources. In particular, a large amount of memory is
required by commonly used fully-connected layers, making it hard to use the
models on low-end devices and stopping the further increase of the model size.
In this paper we convert the dense weight matrices of the fully-connected
layers to the Tensor Train format such that the number of parameters is reduced
by a huge factor and at the same time the expressive power of the layer is
preserved. In particular, for the Very Deep VGG networks we report the
compression factor of the dense weight matrix of a fully-connected layer up to
200000 times leading to the compression factor of the whole network up to 7
times.

本论文研究了如何使用 Tensor Train 格式转换全连接层的矩阵从而减少深度神经网络的参数数量和计算量，试验结论表明该方法可以将全连接层的参数压缩两百万倍，对整个网络的压缩比可达 7 倍。