Progress in the application of machine learning techniques to the prediction of solid-state and molecular materials properties has been greatly facilitated by the development state-of-the-art feature representations and novel deep learning architectures. A large class of atomic structure representations based on expansions of smoothed atomic densities have been shown to correspond to specific choices of basis sets in an abstract many-body Hilbert space. Concurrently, tensor network structures, conventionally the purview of quantum many-body physics and quantum information, have been successfully applied in supervised and unsupervised learning tasks in computer vision and natural language processing. In this work, we argue that architectures based on tensor networks are well-suited to machine learning on Hilbert-space representations of atomic structures. This is demonstrated on supervised learning tasks involving widely available datasets of density functional theory calculations of metal and semiconductor alloys. In particular, we show that certain standard tensor network topologies exhibit strong generalizability even on small training datasets while being parametrically efficient. We further relate this generalizability to the presence of complex entanglement in the trained tensor networks. We also discuss connections to learning with generalized structural kernels and related strategies for compressing large input feature spaces.
Progress in the application of machine learning techniques to the prediction of solid-state and molecular materials properties has been greatly facilitated by the development state-of-the-art feature representations and novel deep learning architectures. A large class of atomic structure representations based on expansions of smoothed atomic densities have been shown to correspond to specific choices of basis sets in an abstract many-body Hilbert space. Concurrently, tensor network structures, conventionally the purview of quantum many-body physics and quantum information, have been successfully applied in supervised and unsupervised learning tasks in computer vision and natural language processing. In this work, we argue that architectures based on tensor networks are well-suited to machine learning on Hilbert-space representations of atomic structures. This is demonstrated on supervised learning tasks involving widely available datasets of density functional theory calculations of metal and semiconductor alloys. In particular, we show that certain standard tensor network topologies exhibit strong generalizability even on small training datasets while being parametrically efficient. We further relate this generalizability to the presence of complex entanglement in the trained tensor networks. We also discuss connections to learning with generalized structural kernels and related strategies for compressing large input feature spaces.
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.
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⟩ (built as a superposition of product states corresponding to images in the training set) and investigate its entanglement properties. We conclude that ∣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×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.
The resemblance between the methods used in studying quantum-many body physics and in machine learning has drawn considerable attention. In particular, tensor networks (TNs) and deep learning architectures bear striking similarities to the extent that TNs can be used for machine learning. Previous results used one-dimensional TNs in image recognition, showing limited scalability and a high bond dimension. In this work, we train two-dimensional hierarchical TNs to solve image recognition problems, using a training algorithm derived from the multipartite entanglement renormalization ansatz (MERA). This approach overcomes scalability issues and implies novel mathematical connections among quantum many-body physics, quantum information theory, and machine learning. While keeping the TN unitary in the training phase, TN states can be defined, which optimally encodes each class of the images into a quantum many-body state. We study the quantum features of the TN states, including quantum entanglement and fidelity. We suggest these quantities could be novel properties that characterize the image classes, as well as the machine learning tasks. Our work could be further applied to identifying possible quantum properties of certain artificial intelligence methods.
Tensor networks were developed in the context of many-body physics as compressed representations of multiparticle quantum states. These representations mitigate the exponential complexity of many-body systems by capturing only the most relevant dependencies. Due to the formal similarity between quantum entanglement and statistical correlations, tensor networks have recently been integrated in machine learning, operating both as alternative learning architectures and as decompositions of components of neural networks. The expectation is that the theoretical understanding of tensor networks developed within quantum many-body physics leads to novel methods that offer advantages in terms of computational efficiency, explainability, or privacy. Here we review the use of tensor networks in the context of machine learning, providing a critical assessment of the state of the art, the potential advantages, and the challenges that must be overcome.
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 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. The models we consider also have a natural implementation on a quantum computer and may guide the development of near-term quantum machine learning architectures.
Abstract The resemblance between the methods used in quantum-many body physics and in machine learning has drawn considerable attention. In particular, tensor networks (TNs) and deep learning architectures bear striking similarities to the extent that TNs can be used for machine learning. Previous results used one-dimensional TNs in image recognition, showing limited scalability and flexibilities. In this work, we train two-dimensional hierarchical TNs to solve image recognition problems, using a training algorithm derived from the multi-scale entanglement renormalization ansatz. This approach introduces mathematical connections among quantum many-body physics, quantum information theory, and machine learning. While keeping the TN unitary in the training phase, TN states are defined, which encode classes of images into quantum many-body states. We study the quantum features of the TN states, including quantum entanglement and fidelity. We find these quantities could be properties that characterize the image classes, as well as the machine learning tasks.