Tensor networks approximate order-$N$ tensors with a reduced number of
degrees of freedom that is only polynomial in $N$ and arranged as a network of
partially contracted smaller tensors. As suggested in [arXiv:2205.15296] in the
context of quantum many-body physics, computation costs can be further
substantially reduced by imposing constraints on the canonical polyadic (CP)
rank of the tensors in such networks. Here we demonstrate how tree tensor
networks (TTN) with CP rank constraints and tensor dropout can be used in
machine learning. The approach is found to outperform other tensor-network
based methods in Fashion-MNIST image classification. A low-rank TTN classifier
with branching ratio $b=4$ reaches test set accuracy 90.3\% with low
computation costs. Consisting of mostly linear elements, tensor network
classifiers avoid the vanishing gradient problem of deep neural networks. The
CP rank constraints have additional advantages: The number of parameters can be
decreased and tuned more freely to control overfitting, improve generalization
properties, and reduce computation costs. They allow us to employ trees with
large branching ratios which substantially improves the representation power.

本文介绍如何将带有 CP 秩约束和张量丢弃功能的树张量网络应用于机器学习中，该方法在 Fashion-MNIST 图像分类中优于其他基于张量网络的方法，且分类器的参数数量可以减少并更灵活地进行控制。

利用树形张量网络、CP 秩约束和张量 Dropout 进行机器学习

Machine learning with tree tensor networks, CP rank constraints, and  tensor dropout

Inspired by coarse-graining approaches used in physics, we show how similar
algorithms can be adapted for data. The resulting algorithms are based on
layered tree tensor networks and scale linearly with both the dimension of the
input and the training set size. Computing most of the layers with an
unsupervised algorithm, then optimizing just the top layer for supervised
classification of the MNIST and fashion-MNIST data sets gives very good
results. We also discuss mixing a prior guess for supervised weights together
with an unsupervised representation of the data, yielding a smaller number of
features nevertheless able to give good performance.

本文介绍了受物理中粗粒化方法启发的相似算法如何适用于数据，并基于分层树张量网络，在输入维度和数据集大小的线性比例上进行缩放，使用无监督算法计算大多数层，然后仅针对 MNIST 和 fashion-MNIST 数据集的顶层进行监督分类的优化，得到了非常好的结果，并讨论了将监督权重的先验猜测与数据的无监督表示相混合，从而得到数量较少的特征，但依然能够获得良好性能。