Unsupervised Generative Modeling Using Matrix Product States

Generative modeling, which learns joint probability distribution from data and generates samples according to it, is an important task in machine learning and artificial intelligence. Inspired by probabilistic interpretation of quantum physics, we propose a generative model using matrix product states, which is a tensor network originally proposed for describing (particularly one-dimensional) entangled quantum states. Our model enjoys efficient learning analogous to the density matrix renormalization group method, which allows dynamically adjusting dimensions of the tensors and offers an efficient direct sampling approach for generative tasks. We apply our method to generative modeling of several standard data sets including the Bars and Stripes random binary patterns and the MNIST handwritten digits to illustrate the abilities, features, and drawbacks of our model over popular generative models such as the Hopfield model, Boltzmann machines, and generative adversarial networks. Our work sheds light on many interesting directions of future exploration in the development of quantum-inspired algorithms for unsupervised machine learning, which are promisingly possible to realize on quantum devices.

Popular Summary

Given a set of statistical data, how could one construct a model that is able to describe the underlying probability distribution and to generate new samples from that distribution? Generative modeling, one of the central tasks in machine learning, provides a method. Quantum generative modeling, exploiting the inherently probabilistic nature of quantum states and their superior representation power, is currently being explored as one of the quantum machine learning approaches. Here, we develop a quantum generative model and examine its abilities on several standard data sets.

Our model encodes a probability distribution into the squared norm of a matrix (tensor) product state (MPS), a potent representation for entangled quantum states. The components of the tensors in the matrix product state are then the parameters that the training algorithm in the generative modeling process must determine from the data. This quantum-mechanics-inspired representation allows us to combine methods from both quantum physics and machine learning to gain advantages in both training and sampling. The training algorithm not only tunes the values of the model parameters but also adaptively adjusts their allocations in the tensors—a flexibility enabled by the MPS tensor network representation. Our model also has a direct sampling algorithm that generates independent samples more efficiently than the conventional statistical-physics-inspired approaches.

Probabilistic modeling using quantum states, bringing together ideas from machine learning and quantum physics, has the potential to be adopted in several areas such as quantum information, quantum computing, and many-body physics. Perhaps the most exciting possibility is the implementation of a quantum generative model in an actual quantum device rather than its simulation on a classical computer. Our approach is a step towards this goal.

Figure 1

(a) The Bars and Stripes data set. (b) Ordering of the pixels when transforming the image into a one-dimensional vector. The numbers between pixels indicate the bond dimensions of the well-trained MPS.

Figure 2

NLL averaged as a function of (a) number of random patterns used for training, with system size $N=20$. (b) System size $N$, trained using $|\mathcal{T}|=100$ random patterns. In both (a) and (b), different symbols correspond to different values of maximal bond dimension ${\mathcal{D}}_{\max }$. Each data point is averaged over 10 random instances (i.e., sets of random patterns); error bars are also plotted, although they are much smaller than symbol size. The black dashed lines in figures denote $\mathcal{L}=\ln |\mathcal{T}|$.

Figure 3

NLL average of a MPS trained using $|\mathcal{T}|=1000$ MNIST images of size $28\times 28$, with varying maximum bond dimensions ${\mathcal{D}}_{\max }$. The horizontal dashed line indicates the Shannon entropy of the training set $\ln |\mathcal{T}|$, which is also the minimal value of $\mathcal{L}$. The inset images are generated by the MPS trained with different ${\mathcal{D}}_{\max }$ (denoted by the arrows).

Figure 4

Bond dimensions of the MPS trained with $|\mathcal{T}|=1000$ MNIST samples, constrained to ${\mathcal{D}}_{\max }=800$. Final average NLL reaches 16.8. Each pixel in this figure corresponds to the bond dimension of the right leg of the tensor associated to the identical coordinate in the original image.

Figure 5

(a) Images generated from the same MPS as in Fig. 4. (b) Original images randomly selected from the training set.

Figure 6

Image reconstruction from partial images by direct sampling with the same MPS as in Fig. 4. (a,b) Restoration of images in Fig. 5, which are selected from the training set. (c,d) Reconstruction of 16 images chosen from the test set. The test set contains images from the MNIST database that were not used for training. The given parts are in black (dark) and the reconstructed parts are in yellow (light). The reconstructed parts are 12 columns from either (a,c) the left or the right and (b,d) the top or the bottom.

Figure 7

Evolution of the average negative log-likelihood $\mathcal{L}$ for both training images (blue, bottom lines) and $10^4$ test images (red, top lines) during training. From left to right, the numbers of images in the training set $|\mathcal{T}|$ are $10^3$, ${10}^4$, and $6\times {10}^4$, respectively.

Figure 8

Top: Distribution of $-\ln p$ of 60 000 training images and 10 000 test images given by a trained MPS with ${\mathcal{D}}_{\max }=500$. The training negative log likelihood ${\mathcal{L}}_{\text{train}}=24.2$, and the test ${\mathcal{L}}_{\mathrm{test}}=30.3$. Bottom: Distributions for each digit.