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.

翻译：张量网络通过减少自由度来近似N阶张量，其自由度数量仅为N的多项式阶，并以部分缩并的小型张量网络形式排列。正如量子多体物理领域文献[arXiv:2205.15296]所述，通过对该网络中张量的规范多面体(CP)秩施加约束，可进一步大幅降低计算成本。本文展示了如何将具有CP秩约束和张量dropout的树张量网络(TTN)应用于机器学习。实验表明，该方法在Fashion-MNIST图像分类任务中优于其他基于张量网络的方法。采用分支比$b=4$的低秩TTN分类器，在较低计算成本下达到90.3%的测试集准确率。由于主要由线性元件构成，张量网络分类器避免了深度神经网络的梯度消失问题。CP秩约束还具有额外优势：可减少参数量并更灵活地调节参数，从而控制过拟合、改善泛化性能并降低计算成本。该约束使我们能够采用大分支比的树结构，从而显著提升表征能力。