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arXiv:2009.09932 · Physical review. B./Physical review. B

Supervised learning with projected entangled pair states

Song Cheng, Lei Wang, Pan Zhang

Published Mar 8, 2021• 59 citations
arXivHTMLPDF

Abstract

Tensor networks, a model that originated from quantum physics, has been gradually generalized as efficient models in machine learning in recent years. However, in order to achieve exact contraction, only treelike tensor networks such as the matrix product states and tree tensor networks have been considered, even for modeling two-dimensional data such as images. In this work, we construct supervised learning models for images using the projected entangled pair states (PEPS), a two-dimensional tensor network having a similar structure prior to natural images. Our approach first performs a feature map, which transforms the image data to a product state on a grid, then contracts the product state to a PEPS with trainable parameters to predict image labels. The tensor elements of PEPS are trained by minimizing differences between training labels and predicted labels. The proposed model is evaluated on image classifications using the Modified National Institute of Standards and Technology database (MNIST) and the Fashion-MNIST datasets. We show that our model is significantly superior to existing models using treelike tensor networks. Moreover, using the same input features, our method performs as well as the multilayer perceptron classifier, but with much fewer parameters and is more stable. Our results shed light on potential applications of two-dimensional tensor network models in machine learning.

DOI
10.1103/physrevb.103.125117

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Tensor networks, a model that originated from quantum physics, has been gradually generalized as efficient models in machine learning in recent years. However, in order to achieve exact contraction, only treelike tensor networks such as the matrix product states and tree tensor networks have been considered, even for modeling two-dimensional data such as images. In this work, we construct supervised learning models for images using the projected entangled pair states (PEPS), a two-dimensional tensor network having a similar structure prior to natural images. Our approach first performs a feature map, which transforms the image data to a product state on a grid, then contracts the product state to a PEPS with trainable parameters to predict image labels. The tensor elements of PEPS are trained by minimizing differences between training labels and predicted labels. The proposed model is evaluated on image classifications using the Modified National Institute of Standards and Technology database (MNIST) and the Fashion-MNIST datasets. We show that our model is significantly superior to existing models using treelike tensor networks. Moreover, using the same input features, our method performs as well as the multilayer perceptron classifier, but with much fewer parameters and is more stable. Our results shed light on potential applications of two-dimensional tensor network models in machine learning.
Tensor networks have been recognized as a powerful numerical tool; they are applied in various fields, including physics, computer science, and more. The idea of a tensor network originates from quantum physics as an efficient representation of quantum many-body states and their operations. Matrix product states (MPS) form one of the simplest tensor networks and have been applied to machine learning for image classification. However, MPS has certain limitations when processing two-dimensional images, meaning that it is preferable for an projected entangled pair states (PEPS) tensor network with a similar structure to the image to be introduced into machine learning. PEPS tensor networks are significantly superior to other tensor networks on the image classification task. Based on a PEPS tensor network, this paper constructs a multi-layered PEPS (MLPEPS) tensor network model for image classification. PEPS is used to extract features layer by layer from the image mapped to the Hilbert space, which fully utilizes the correlation between pixels while retaining the global structural information of the image. When performing classification tasks on the Fashion-MNIST dataset, MLPEPS achieves a classification accuracy of 90.44%, exceeding tensor network models such as the original PEPS. On the COVID-19 radiography dataset, MLPEPS has a test set accuracy of 91.63%, which is very close to the results of GoogLeNet. Under the same experimental conditions, the learning ability of MLPEPS is already close to that of existing neural networks while having fewer parameters. MLPEPS can be used to build different network models by modifying the structure, and as such it has great potential in machine learning.
Tensor networks, originally designed to address computational problems in quantum many-body physics, have recently been applied to machine learning tasks. However, compared to quantum physics, where the reasons for the success of tensor network approaches over the last 30 years is well understood, very little is yet known about why these techniques work for machine learning. The goal of this paper is to investigate entanglement properties of tensor network models in a current machine learning application, in order to uncover general principles that may guide future developments. We revisit the use of tensor networks for supervised image classification using the MNIST data set of handwritten digits, as pioneered by Stoudenmire and Schwab [Adv. in Neur. Inform. Proc. Sys. 29, 4799 (2016)]. Firstly we hypothesize about which state the tensor network might be learning during training. For that purpose, we propose a plausible candidate state ∣Σℓ⟩|Σ_{\ell}\rangle∣Σℓ​⟩ (built as a superposition of product states corresponding to images in the training set) and investigate its entanglement properties. We conclude that ∣Σℓ⟩|Σ_{\ell}\rangle∣Σℓ​⟩ is so robustly entangled that it cannot be approximated by the tensor network used in that work, which must therefore be representing a very different state. Secondly, we use tensor networks with a block product structure, in which entanglement is restricted within small blocks of n×nn \times nn×n pixels/qubits. We find that these states are extremely expressive (e.g. training accuracy of 99.97%99.97 \%99.97% already for n=2n=2n=2), suggesting that long-range entanglement may not be essential for image classification. However, in our current implementation, optimization leads to over-fitting, resulting in test accuracies that are not competitive with other current approaches.
Tensor networks are efficient representations of high-dimensional tensors which have been very successful for physics and mathematics applications. We demonstrate how algorithms for optimizing such networks can be adapted to supervised learning tasks by using matrix product states (tensor trains) to parameterize models for classifying images. For the MNIST data set we obtain less than 1% test set classification error. We discuss how the tensor network form imparts additional structure to the learned model and suggest a possible generative interpretation.
Matrix product states (MPSs), a tensor network designed for one-dimensional quantum systems, were recently proposed for generative modeling of natural data (such as images) in terms of the ``Born machine.'' However, the exponential decay of correlation in MPSs restricts its representation power heavily for modeling complex data such as natural images. In this work, we push forward the effort of applying tensor networks to machine learning by employing the tree tensor network (TTN), which exhibits balanced performance in expressibility and efficient training and sampling. We design the tree tensor network to utilize the two-dimensional prior of the natural images and develop sweeping learning and sampling algorithms which can be efficiently implemented utilizing graphical processing units. We apply our model to random binary patterns and the binary MNIST data sets of handwritten digits. We show that the TTN is superior to MPSs for generative modeling in keeping the correlation of pixels in natural images, as well as giving better log-likelihood scores in standard data sets of handwritten digits. We also compare its performance with state-of-the-art generative models such as variational autoencoders, restricted Boltzmann machines, and PixelCNN. Finally, we discuss the future development of tensor network states in machine learning problems.
The tensor network (TN) has recently triggered extensive interests in developing machine learning models defined in quantum Hilbert space. Here, we propose a generative TN classification (GTNC) model for supervised machine learning. The strategy is first to map the classical data onto the states in a many-body Hilbert space and then to capture these states with tensor network schemes. We adopt the TN in the form of matrix product states as an example to implement GTNC where the testing images are classified by comparing the fidelities between different states. Our results show that GTNC has a very impressive performance on benchmark datasets in comparison to several well-known machine learning models. The advantage of GTNC is reflected from the facts that the samples are naturally clustering in the many-body Hilbert space, and it relies much less on hyperparameters. These characters make GTNC an adaptive and universal quantum-inspired method, which would have important applications in quantum computation and quantum information.
We propose a restricted class of tensor network state, built from number-state preserving tensors, for supervised learning tasks. This class of tensor network is argued to be a natural choice for classifiers as 1) they map classical data to classical data, and thus preserve the interpretability of data under tensor transformations, 2) they can be efficiently trained to maximize their scalar product against classical data sets, and 3) they seem to be as powerful as generic (unrestricted) tensor networks in this task. Our proposal is demonstrated using a variety of benchmark classification problems, where number-state preserving versions of commonly used networks (including MPS, TTN and MERA) are trained as effective classifiers. This work opens the path for powerful tensor network methods such as MERA, which were previously computationally intractable as classifiers, to be employed for difficult tasks such as image recognition.