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arXiv:2103.06872 · Physical Review A

Tensor networks and efficient descriptions of classical data

Sirui Lu, Márton Kanász-Nagy, Ivan Kukuljan, J. I. Cirac

Published Mar 6, 2025• 13 citations
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Abstract

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.

DOI
10.1103/physreva.111.032409

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•Oct 13, 2017•13 citations
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 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.
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 (TN) are approximations of high-dimensional tensors designed to represent locally entangled quantum many-body systems efficiently. This paper provides a comprehensive comparison between classical TNs and TN-inspired quantum circuits in the context of machine learning on highly complex, simulated Large Hadron Collider data. We show that classical TNs require exponentially large bond dimensions and higher Hilbert-space mapping to perform comparably to their quantum counterparts. While such an expansion in the dimensionality allows better performance, we observe that, with increased dimensionality, classical TNs lead to a highly flat loss landscape, rendering the usage of gradient-based optimization methods highly challenging. Furthermore, by employing quantitative metrics, such as the Fisher information and effective dimensions, we show that classical TNs require a more extensive training sample to represent the data as efficiently as TN-inspired quantum circuits. We also engage with the idea of hybrid classical-quantum TNs and show possible architectures to employ a larger phase space from the data. We offer our results using three main TN Ans\"atze: Tree tensor networks, matrix product states, and multiscale entanglement renormalization Ans\"atze.
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.
Tensor network states are an indispensable tool for the simulation of strongly correlated quantum many-body systems. In recent years, tree tensor network states (TTNS) have been successfully used for two-dimensional systems and to benchmark quantum simulation approaches for condensed matter, nuclear, and particle physics. In comparison to the more traditional approach based on matrix product states (MPS), the graph distance of physical degrees of freedom can be drastically reduced in TTNS. Surprisingly, it turns out that, for large systems in D>1D>1D>1 spatial dimensions, MPS simulations of low-energy states are nevertheless more efficient than TTNS simulations. With a focus on D=2D=2D=2 and 3, the scaling of computational costs for different boundary conditions is determined under the assumption that the system obeys an entanglement (log-)area law, implying that bond dimensions scale exponentially in the surface area of the associated subsystems.
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.