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arXiv:2007.06082

Entanglement and Tensor Networks for Supervised Image Classification

John M. Martyn, Guifré Vidal, Chase Roberts, Stefan Leichenauer

Published Jul 12, 2020• 23 citations
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Abstract

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 $|Σ_{\ell}\rangle$ (built as a superposition of product states corresponding to images in the training set) and investigate its entanglement properties. We conclude that $|Σ_{\ell}\rangle$ 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 $n \times n$ pixels/qubits. We find that these states are extremely expressive (e.g. training accuracy of $99.97 \%$ already for $n=2$), 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.

DOI
10.48550/arxiv.2007.06082

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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.
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 are approximations of high-order tensors which are efficient to work with and have been very successful for physics and mathematics applications. We demonstrate how algorithms for optimizing tensor networks can be adapted to supervised learning tasks by using matrix product states (tensor trains) to parameterize non-linear kernel learning models. For the MNIST data set we obtain less than 1% test set classification error. We discuss an interpretation of the additional structure imparted by the tensor network to the learned model.
Machine learning (ML) has recently facilitated many advances in solving problems related to many-body physical systems. Given the intrinsic quantum nature of these problems, it is natural to speculate that quantum-enhanced machine learning will enable us to unveil even greater details than we currently have. With this motivation, this paper examines a quantum machine learning approach based on shallow variational ansatz inspired by tensor networks for supervised learning tasks. In particular, we first look at the standard image classification tasks using the Fashion-MNIST dataset and study the effect of repeating tensor network layers on ansatz's expressibility and performance. Finally, we use this strategy to tackle the problem of quantum phase recognition for the transverse-field Ising and Heisenberg spin models in one and two dimensions, where we were able to reach ≥98%\geq 98\%≥98% test-set accuracies with both multi-scale entanglement renormalization ansatz (MERA) and tree tensor network (TTN) inspired parametrized quantum circuits.
Tensor networks have found a wide use in a variety of applications in physics and computer science, recently leading to both theoretical insights as well as practical algorithms in machine learning. In this work we explore the connection between tensor networks and probabilistic graphical models, and show that it motivates the definition of generalized tensor networks where information from a tensor can be copied and reused in other parts of the network. We discuss the relationship between generalized tensor network architectures used in quantum physics, such as String-Bond States and Entangled Plaquette States, and architectures commonly used in machine learning. We provide an algorithm to train these networks in a supervised learning context and show that they overcome the limitations of regular tensor networks in higher dimensions, while keeping the computation efficient. A method to combine neural networks and tensor networks as part of a common deep learning architecture is also introduced. We benchmark our algorithm for several generalized tensor network architectures on the task of classifying images and sounds, and show that they outperform previously introduced tensor network algorithms. Some of the models we consider can be realized on a quantum computer and may guide the development of near-term quantum machine learning architectures.
Tiny object classification problem exists in many machine learning applications like medical imaging or remote sensing, where the object of interest usually occupies a small region of the whole image. It is challenging to design an efficient machine learning model with respect to tiny object of interest. Current neural network structures are unable to deal with tiny object efficiently because they are mainly developed for images featured by large scale objects. However, in quantum physics, there is a great theoretical foundation guiding us to analyze the target function for image classification regarding to specific objects size ratio. In our work, we apply Tensor Networks to solve this arising tough machine learning problem. First, we summarize the previous work that connects quantum spin model to image classification and bring the theory into the scenario of tiny object classification. Second, we propose using 2D multi-scale entanglement renormalization ansatz (MERA) to classify tiny objects in image. In the end, our experimental results indicate that tensor network models are effective for tiny object classification problem and potentially will beat state-of-the-art. Our codes will be available online https://github.com/timqqt/MERA_Image_Classification.