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arXiv:2009.12280 · The Journal of Machine Learning for Biomedical Imaging

Locally orderless tensor networks for classifying two- and three-dimensional medical images

Raghavendra Selvan, Silas Nyboe Ørting, Erik B. Dam

Published Mar 23, 2021• 6 citations
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Abstract

Tensor networks are factorisations of high rank tensors into networks of lower rank tensors and have primarily been used to analyse quantum many-body problems. Tensor networks have seen a recent surge of interest in relation to supervised learning tasks with a focus on image classification. In this work, we improve upon the matrix product state (MPS) tensor networks that can operate on one-dimensional vectors to be useful for working with 2D and 3D medical images. We treat small image regions as orderless, squeeze their spatial information into feature dimensions and then perform MPS operations on these locally orderless regions. These local representations are then aggregated in a hierarchical manner to retain global structure. The proposed locally orderless tensor network (LoTeNet) is compared with relevant methods on three datasets. The architecture of LoTeNet is fixed in all experiments and we show it requires lesser computational resources to attain performance on par or superior to the compared methods.

DOI
10.59275/j.melba.2021-g65b

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Tensor networks are factorisations of high rank tensors into networks of lower rank tensors and have primarily been used to analyse quantum many-body problems. Tensor networks have seen a recent surge of interest in relation to supervised learning tasks with a focus on image classification. In this work, we improve upon the matrix product state (MPS) tensor networks that can operate on one-dimensional vectors to be useful for working with 2D and 3D medical images. We treat small image regions as orderless, squeeze their spatial information into feature dimensions and then perform MPS operations on these locally orderless regions. These local representations are then aggregated in a hierarchical manner to retain global structure. The proposed locally orderless tensor network (LoTeNet) is compared with relevant methods on three datasets. The architecture of LoTeNet is fixed in all experiments and we show it requires lesser computational resources to attain performance on par or superior to the compared methods.
With the increasing adoption of machine learning tools like neural networks across several domains, interesting connections and comparisons to concepts from other domains are coming to light. In this work, we focus on the class of Tensor Networks, which has been a work horse for physicists in the last two decades to analyse quantum many-body systems. Building on the recent interest in tensor networks for machine learning, we extend the Matrix Product State tensor networks (which can be interpreted as linear classifiers operating in exponentially high dimensional spaces) to be useful in medical image analysis tasks. We focus on classification problems as a first step where we motivate the use of tensor networks and propose adaptions for 2D images using classical image domain concepts such as local orderlessness of images. With the proposed locally orderless tensor network model (LoTeNet), we show that tensor networks are capable of attaining performance that is comparable to state-of-the-art deep learning methods. We evaluate the model on two publicly available medical imaging datasets and show performance improvements with fewer model hyperparameters and lesser computational resources compared to relevant baseline methods.
The recently introduced locally orderless tensor network (LoTeNet) for supervised image classification uses matrix product state (MPS) operations on grids of transformed image patches. The resulting patch representations are combined back together into the image space and aggregated hierarchically using multiple MPS blocks per layer to obtain the final decision rules. In this work, we propose a non-patch based modification to LoTeNet that performs one MPS operation per layer, instead of several patch-level operations. The spatial information in the input images to MPS blocks at each layer is squeezed into the feature dimension, similar to LoTeNet, to maximise retained spatial correlation between pixels when images are flattened into 1D vectors. The proposed multi-layered tensor network (MLTN) is capable of learning linear decision boundaries in high dimensional spaces in a multi-layered setting, which results in a reduction in the computation cost compared to LoTeNet without any degradation in performance.
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 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.
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.
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.