Tensor networks for unsupervised machine learning

Modeling the joint distribution of high-dimensional data is a central task in unsupervised machine learning. In recent years, many interests have been attracted to developing learning models based on tensor networks, which have the advantages of a principle understanding of the expressive power using entanglement properties, and as a bridge connecting classical computation and quantum computation. Despite the great potential, however, existing tensor network models for unsupervised machine learning only work as a proof of principle, as their performance is much worse than the standard models such as restricted Boltzmann machines and neural networks. In this Letter, we present autoregressive matrix product states (AMPS), a tensor network model combining matrix product states from quantum many-body physics and autoregressive modeling from machine learning. Our model enjoys the exact calculation of normalized probability and unbiased sampling. We demonstrate the performance of our model using two applications, generative modeling on synthetic and real-world data, and reinforcement learning in statistical physics. Using extensive numerical experiments, we show that the proposed model significantly outperforms the existing tensor network models and the restricted Boltzmann machines, and is competitive with state-of-the-art neural network models.

Figure 1

Schematic representation of generative models based on (a) MPS, (b) TTN, (c) RBM with a two-dimensional tensor network representation [18], and (d) AMPS. In (d), the joint probability is factorized as the product of conditional probabilities and each of them is parametrized using an MPS of a different length. In (a), (b), and (d), the solid box denotes the nonlinear or linear function to normalize the tensor network.

Figure 2

(a) For random data sets with 20 variables, we plot the NLL vs the number of samples $\left|\mathcal{T}\right|$ for AMPS, MPS Born machine with real entries ($\text{MPS}{\text{BM}}_{\mathbb{R}}$), and RBM. We also give the number of parameters of each model in the legend. (b) For the real-world data set called Lymphography [33], we plot the NLL as a function of bond dimension $D$ for AMPS, positive MPS (${\text{MPS}}_{\mathbb{R}\ge 0}$), MPS Born machines with real entries ($\text{MPS}{\text{BM}}_{\mathbb{R}}$) and complex entries ($\text{MPS}{\text{BM}}_{\mathbb{C}}$), and locally purified states with real entries (${\text{LPS}}_{\mathbb{R}}$) and complex entries (${\text{LPS}}_{\mathbb{C}}$). The dashed line indicates the lower bound and is determined by the entropy of the empirical data distribution.

Figure 3

(a) Handwritten digits in the binarized MNIST data set. (b) Digits sampled from the learned AMPS ($D=100$), $\text{test}\text{NLL}=84.1$. (c) Digits sampled from the learned Deep-AMPS (8 layers and $D=14$), test $\mathrm{NLL}=81.8$.

Figure 4

Free energy given by the naive mean field (NMF) [46], variational autoregressive network (VAN) [19], and the AMPS on the SK spin-glass model [45] with 20 spins. The inset plot shows the relative error on a logarithmic scale. In the experiment, the VAN (1 layer) and the AMPS ($D=2$) have roughly the same number of parameters.