Gang Su

Shi-Ju Ran, Gang Su

It is a critical challenge to simultaneously gain high interpretability and efficiency with the current schemes of deep machine learning (ML). Tensor network (TN), which is a well-established mathematical tool originating from quantum mechanics, has shown its unique advantages on developing efficient ``white-box'' ML schemes. Here, we give a brief review on the inspiring progresses made in TN-based ML. On one hand, interpretability of TN ML is accommodated with the solid theoretical foundation based on quantum information and many-body physics. On the other hand, high efficiency can be rendered from the powerful TN representations and the advanced computational techniques developed in quantum many-body physics. With the fast development on quantum computers, TN is expected to conceive novel schemes runnable on quantum hardware, heading towards the ``quantum artificial intelligence'' in the forthcoming future.

Zheng-zhi Sun, Shi-ju Ran, Gang Su

The gradient-based optimization method for deep machine learning models suffers from gradient vanishing and exploding problems, particularly when the computational graph becomes deep. In this work, we propose the tangent-space gradient optimization (TSGO) for the probabilistic models to keep the gradients from vanishing or exploding. The central idea is to guarantee the orthogonality between the variational parameters and the gradients. The optimization is then implemented by rotating parameter vector towards the direction of gradient. We explain and testify TSGO in tensor network (TN) machine learning, where the TN describes the joint probability distribution as a normalized state $\left| ψ\right\rangle $ in Hilbert space. We show that the gradient can be restricted in the tangent space of $\left\langle ψ\right.\left| ψ\right\rangle = 1$ hyper-sphere. Instead of additional adaptive methods to control the learning rate in deep learning, the learning rate of TSGO is naturally determined by the angle $θ$ as $η= \tan θ$. Our numerical results reveal better convergence of TSGO in comparison to the off-the-shelf Adam.

Zheng-Zhi Sun, Cheng Peng, Ding Liu et al.

Tensor network (TN) has recently triggered extensive interests in developing machine-learning models in quantum many-body Hilbert space. Here we purpose a generative TN classification (GTNC) approach for supervised learning. The strategy is to train the generative TN for each class of the samples to construct the classifiers. The classification is implemented by comparing the distance in the many-body Hilbert space. The numerical experiments by GTNC show impressive performance on the MNIST and Fashion-MNIST dataset. The testing accuracy is competitive to the state-of-the-art convolutional neural network while higher than the naive Bayes classifier (a generative classifier) and support vector machine. Moreover, GTNC is more efficient than the existing TN models that are in general discriminative. By investigating the distances in the many-body Hilbert space, we find that (a) the samples are naturally clustering in such a space; and (b) bounding the bond dimensions of the TN's to finite values corresponds to removing redundant information in the image recognition. These two characters make GTNC an adaptive and universal model of excellent performance.