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arXiv:2103.00105 · Machine Learning Science and Technology

Mutual information scaling for tensor network machine learning

Ian Convy, William J. Huggins, Haoran Liao, K. Birgitta Whaley

Published Dec 20, 2021• 20 citations
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Abstract

Tensor networks have emerged as promising tools for machine learning, inspired by their widespread use as variational ansatze in quantum many-body physics. It is well known that the success of a given tensor network ansatz depends in part on how well it can reproduce the underlying entanglement structure of the target state, with different network designs favoring different scaling patterns. We demonstrate here how a related correlation analysis can be applied to tensor network machine learning, and explore whether classical data possess correlation scaling patterns similar to those found in quantum states which might indicate the best network to use for a given dataset. We utilize mutual information as measure of correlations in classical data, and show that it can serve as a lower-bound on the entanglement needed for a probabilistic tensor network classifier. We then develop a logistic regression algorithm to estimate the mutual information between bipartitions of data features, and verify its accuracy on a set of Gaussian distributions designed to mimic different correlation patterns. Using this algorithm, we characterize the scaling patterns in the MNIST and Tiny Images datasets, and find clear evidence of boundary-law scaling in the latter. This quantum-inspired classical analysis offers insight into the design of tensor networks which are best suited for specific learning tasks.

DOI
10.1088/2632-2153/ac44a9

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Tensor networks have emerged as promising tools for machine learning, inspired by their widespread use as variational ansatze in quantum many-body physics. It is well known that the success of a given tensor network ansatz depends in part on how well it can reproduce the underlying entanglement structure of the target state, with different network designs favoring different scaling patterns. We demonstrate here how a related correlation analysis can be applied to tensor network machine learning, and explore whether classical data possess correlation scaling patterns similar to those found in quantum states which might indicate the best network to use for a given dataset. We utilize mutual information as measure of correlations in classical data, and show that it can serve as a lower-bound on the entanglement needed for a probabilistic tensor network classifier. We then develop a logistic regression algorithm to estimate the mutual information between bipartitions of data features, and verify its accuracy on a set of Gaussian distributions designed to mimic different correlation patterns. Using this algorithm, we characterize the scaling patterns in the MNIST and Tiny Images datasets, and find clear evidence of boundary-law scaling in the latter. This quantum-inspired classical analysis offers insight into the design of tensor networks which are best suited for specific learning tasks.
We investigate the potential of tensor-network-based machine-learning methods to scale to large image and text datasets. We study how the mutual information between a subregion and its complement scales with the subsystem size L , similarly to approaches used in quantum many-body physics. Using various mutual information (MI) estimation methods, including an autoregressive network-based estimator, we find that simple image datasets exhibit area law scaling, suggesting efficient representation by two-dimensional tensor networks. More complex image datasets exceed the area law, indicating the need for generalized tensor network states or hybrid models. For Wikipedia text data, we observe power-law scaling I ( L ) ∝ L ν with ν ≈ 0.82 , approaching a volume law. This implies that one-dimensional tensor networks with area law entanglement may not efficiently capture the structure of text. We introduce two models to reproduce this scaling: a quantum-inspired random pair toy model and a linguistically motivated Markovian dependency tree model. In the latter model, matching the observed MI scaling allows us to infer the word-word correlation length distribution in text.
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.
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.
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.
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 network algorithms seek to minimize correlations to compress the classical data representing quantum states. Tensor network algorithms and similar tools—called tensor network methods—form the backbone of modern numerical methods used to simulate many-body physics and have a further range of applications in machine learning. Finding and contracting tensor network states is a computational task, which may be accelerated by quantum computing. We present a quantum algorithm that returns a classical description of a rank-r tensor network state satisfying an area law and approximating an eigenvector given black-box access to a unitary matrix. Our work creates a bridge between several contemporary approaches, including tensor networks, the variational quantum eigensolver (VQE), quantum approximate optimization algorithm (QAOA), and quantum computation.