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arXiv:1901.02217 · Physical review. B./Physical review. B

Tree tensor networks for generative modeling

Song Cheng, Lei Wang, Tao Xiang, Pan Zhang

Published Apr 18, 2019• 123 citations
arXivHTMLPDF

Abstract

Matrix product states (MPSs), a tensor network designed for one-dimensional quantum systems, were recently proposed for generative modeling of natural data (such as images) in terms of the ``Born machine.'' However, the exponential decay of correlation in MPSs restricts its representation power heavily for modeling complex data such as natural images. In this work, we push forward the effort of applying tensor networks to machine learning by employing the tree tensor network (TTN), which exhibits balanced performance in expressibility and efficient training and sampling. We design the tree tensor network to utilize the two-dimensional prior of the natural images and develop sweeping learning and sampling algorithms which can be efficiently implemented utilizing graphical processing units. We apply our model to random binary patterns and the binary MNIST data sets of handwritten digits. We show that the TTN is superior to MPSs for generative modeling in keeping the correlation of pixels in natural images, as well as giving better log-likelihood scores in standard data sets of handwritten digits. We also compare its performance with state-of-the-art generative models such as variational autoencoders, restricted Boltzmann machines, and PixelCNN. Finally, we discuss the future development of tensor network states in machine learning problems.

DOI
10.1103/physrevb.99.155131

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Matrix product states (MPSs), a tensor network designed for one-dimensional quantum systems, were recently proposed for generative modeling of natural data (such as images) in terms of the ``Born machine.'' However, the exponential decay of correlation in MPSs restricts its representation power heavily for modeling complex data such as natural images. In this work, we push forward the effort of applying tensor networks to machine learning by employing the tree tensor network (TTN), which exhibits balanced performance in expressibility and efficient training and sampling. We design the tree tensor network to utilize the two-dimensional prior of the natural images and develop sweeping learning and sampling algorithms which can be efficiently implemented utilizing graphical processing units. We apply our model to random binary patterns and the binary MNIST data sets of handwritten digits. We show that the TTN is superior to MPSs for generative modeling in keeping the correlation of pixels in natural images, as well as giving better log-likelihood scores in standard data sets of handwritten digits. We also compare its performance with state-of-the-art generative models such as variational autoencoders, restricted Boltzmann machines, and PixelCNN. Finally, we discuss the future development of tensor network states in machine learning problems.
Modeling the joint distribution of high-dimensional data is a central task in unsupervised machine learning. In recent years, many interests have been attracted to developing learning models based on tensor networks, which have the advantages of a principle understanding of the expressive power using entanglement properties, and as a bridge connecting classical computation and quantum computation. Despite the great potential, however, existing tensor network models for unsupervised machine learning only work as a proof of principle, as their performance is much worse than the standard models such as restricted Boltzmann machines and neural networks. In this Letter, we present autoregressive matrix product states (AMPS), a tensor network model combining matrix product states from quantum many-body physics and autoregressive modeling from machine learning. Our model enjoys the exact calculation of normalized probability and unbiased sampling. We demonstrate the performance of our model using two applications, generative modeling on synthetic and real-world data, and reinforcement learning in statistical physics. Using extensive numerical experiments, we show that the proposed model significantly outperforms the existing tensor network models and the restricted Boltzmann machines, and is competitive with state-of-the-art neural network models.
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.
We describe a quantum-assisted machine learning method in which multivariate data are encoded into quantum states in a Hilbert space whose dimension is exponentially large in the length of the data vector. Learning in this space occurs through applying a low-depth quantum circuit with a tree-tensor-network (TTN) topology acting as an unsupervised feature extractor to identify the most relevant quantum states in a data-driven fashion and then applying a supervised linear classifier encoding the class decision in a small-dimensional quantum register. We present tools for making TTN classifiers amenable to implementation on gate-based quantum computing devices, including an embedding map with accuracy similar to the recently defined exponential machines (A. Novikov et al., arXiv:1605.03795) but which produces valid quantum state embeddings of classical data vectors, and the use of manifold-based gradient optimization schemes to produce isometric operations mapping quantum states to a register of qubits defining a class decision. We detail methods for efficiently obtaining one-point and two-point correlation functions of the vectors defining the decision boundary of the quantum model, which can be used for model interpretability, as well as methods for obtaining classification decisions from partial data vectors. Further, we show that the use of isometric tensors can significantly aid in the human interpretability of the correlation functions extracted from the decision weights and may produce models that are less susceptible to adversarial perturbations. Finally, we discuss in detail the problem of compiling classically optimized isometric TTN models into unitary operations to be run on quantum computers and how isometric models requiring postselection on quantum hardware can be used to precondition variational Ans\"atze for models without postselection. We demonstrate our methodologies in applications utilizing the MNIST database of handwritten digits and a multivariate time-series data set of human activity recognition.
Quantum machine learning (QML) is a rapidly expanding field that merges the principles of quantum computing with the techniques of machine learning. One of the powerful mathematical frameworks in this domain is tensor networks. These networks are used to approximate high-order tensors by contracting tensors with lower ranks. Initially developed for simulating quantum systems, tensor networks have become integral to quantum computing and, by extension, to QML. Drawing inspiration from these quantum methods, specifically the Matrix Product States (MPS), we apply them in a classical machine learning setting. Their ability to efficiently represent and manipulate complex, high-dimensional data makes them effective in a supervised learning framework. Here, we present an MPS model, in which the MPS functions as both a classifier and a generator. The dual functionality of this novel MPS model permits a strategy that enhances the traditional training of supervised MPS models. This framework is inspired by generative adversarial networks and is geared towards generating more realistic samples by reducing outliers. In addition, our contributions offer insights into the mechanics of tensor network methods for generation tasks. Specifically, we discuss alternative embedding functions and a new sampling method from non-normalized MPSs.
Tensor networks are a powerful modeling framework developed for computational many-body physics, which have only recently been applied within machine learning. In this work we utilize a uniform matrix product state (u-MPS) model for probabilistic modeling of sequence data. We first show that u-MPS enable sequence-level parallelism, with length-n sequences able to be evaluated in depth O(log n). We then introduce a novel generative algorithm giving trained u-MPS the ability to efficiently sample from a wide variety of conditional distributions, each one defined by a regular expression. Special cases of this algorithm correspond to autoregressive and fill-in-the-blank sampling, but more complex regular expressions permit the generation of richly structured text in a manner that has no direct analogue in current generative models. Experiments on synthetic text data find u-MPS outperforming LSTM baselines in several sampling tasks, and demonstrate strong generalization in the presence of limited data.
Tensor networks are a powerful modeling framework developed for computational many-body physics, which have only recently been applied within machine learning. In this work we utilize a uniform matrix product state (u-MPS) model for probabilistic modeling of sequence data. We first show that u-MPS enable sequence-level parallelism, with length-n sequences able to be evaluated in depth O(log n). We then introduce a novel generative algorithm giving trained u-MPS the ability to efficiently sample from a wide variety of conditional distributions, each one defined by a regular expression. Special cases of this algorithm correspond to autoregressive and fill-in-the-blank sampling, but more complex regular expressions permit the generation of richly structured data in a manner that has no direct analogue in neural generative models. Experiments on sequence modeling with synthetic and real text data show u-MPS outperforming a variety of baselines and effectively generalizing their predictions in the presence of limited data.