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arXiv:1710.05520

Entanglement Entropy of Target Functions for Image Classification and Convolutional Neural Network

Zhang, Ya-Hui

Published Oct 16, 2017• 5 citations
arXivHTMLPDF

Abstract

The success of deep convolutional neural network (CNN) in computer vision especially image classification problems requests a new information theory for function of image, instead of image itself. In this article, after establishing a deep mathematical connection between image classification problem and quantum spin model, we propose to use entanglement entropy, a generalization of classical Boltzmann-Shannon entropy, as a powerful tool to characterize the information needed for representation of general function of image. We prove that there is a sub-volume-law bound for entanglement entropy of target functions of reasonable image classification problems. Therefore target functions of image classification only occupy a small subspace of the whole Hilbert space. As a result, a neural network with polynomial number of parameters is efficient for representation of such target functions of image. The concept of entanglement entropy can also be useful to characterize the expressive power of different neural networks. For example, we show that to maintain the same expressive power, number of channels $D$ in a convolutional neural network should scale with the number of convolution layers $n_c$ as $D\sim D_0^{\frac{1}{n_c}}$. Therefore, deeper CNN with large $n_c$ is more efficient than shallow ones.

DOI
10.48550/arxiv.1710.05520

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Convolutional Neural Network (CNN) is a breakthrough learning model that shows outstanding performance in computer vision and deep learning applications. However, it is a relatively burdened model in terms of learning speed and resource usage compared to other learning models when the learning scale becomes large. Quantum Convolutional Neural Network (QCNN) is a novel model as a potential solution using quantum computers to handle this problem. Quantum computers with a limited number of usable qubits needs a resource-efficient method to process large-scale data at once. In addition, Quantum Random Access Memory (QRAM) can store the large data to qubits logarithmically using superposition and entanglement. The QRAM algorithm can design a new QCNN model that can efficiently process in massive data. This paper proposes a more resource and depth efficient model for larger-sized input data and the number of output channels using the QRAM algorithm and efficiently extracting features.
Abstract A study of the artificial neural network representation of quantum many-body states is presented. The locality and entanglement properties of states for shallow and deep quantum neural networks are investigated in detail. By introducing the notion of local quasi-product states, for which the locally connected shallow feed-forward neural network states and restricted Boltzmann machine states are special cases, we show that Rényi entanglement entropies of all these states obey the entanglement area law. Besides, we also investigate the entanglement features of deep Boltzmann machine states and show that locality constraints imposed on the neural networks make the states obey the entanglement area law. Finally, as an application, we apply the notion of Rényi entanglement entropy to understand the power of neural networks, and show that image classification problems can be efficiently solved must obey the area law.
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Convolutional neural networks (CNNs) have been employed along with variational Monte Carlo methods for finding the ground state of quantum many-body spin systems with great success. However, it remains uncertain how CNNs, with a model complexity that scales at most linearly with the number of particles, solve the “curse of dimensionality” and efficiently represent wavefunctions in exponentially large Hilbert spaces. In this work, we use methodologies from information theory, group theory and machine learning, to elucidate how CNN captures relevant physics of quantum systems. We connect CNNs to a class of restricted maximum entropy (MaxEnt) and entangled plaquette correlator product state (EP-CPS) models that approximate symmetry constrained classical correlations between subsystems. For the final part of the puzzle, inspired by similar analyses for matrix product states and tensor networks, we show that the CNNs rely on the spectrum of each subsystem's entanglement Hamiltonians as captured by the size of the convolutional filter. All put together, these allow CNNs to simulate exponential quantum wave functions using a model that scales at most linear in system size as well as provide clues into when CNNs might fail to simulate Hamiltonians. We incorporate our insights into a new training algorithm and demonstrate its improved efficiency, accuracy, and robustness. Finally, we use regression analysis to show how the CNNs solutions can be used to identify salient physical features of the system that are the most relevant to an efficient approximation. Our integrated approach can be extended to similarly analyzing other neural network architectures and quantum spin systems.