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arXiv:1803.11056

Holographic Renormalization with Machine learning

Eric M. Howard

Published Mar 23, 2018• 1 citation
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Abstract

At low energies, the microscopic characteristics and changes of physical systems as viewed at different distance scales are described by universal scale invariant properties investigated by the Renormalization Group (RG) apparatus, an efficient tool used to deal with scaling problems in effective field theories. We employ an information-theoretic approach in a deep learning setup by introducing an artificial neural network algorithm to map and identify new physical degrees of freedom. Using deep learning methods mapped to a genuine field theory, we develop a mechanism capable to identify relevant degrees of freedom and induce scale invariance without prior knowledge about a system. We show that deep learning algorithms that use an RG-like scheme to learn relevant features from data could help to understand the nature of the holographic entanglement entropy and the holographic principle in context of the AdS/CFT correspondence.

DOI
10.48550/arxiv.1803.11056

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At low energies, the microscopic characteristics and changes of physical systems as viewed at different distance scales are described by universal scale invariant properties investigated by the Renormalization Group (RG) apparatus, an efficient tool used to deal with scaling problems in effective field theories. We employ an information-theoretic approach in a deep learning setup by introducing an artificial neural network algorithm to map and identify new physical degrees of freedom. Using deep learning methods mapped to a genuine field theory, we develop a mechanism capable to identify relevant degrees of freedom and induce scale invariance without prior knowledge about a system. We show that deep learning algorithms that use an RG-like scheme to learn relevant features from data could help to understand the nature of the holographic entanglement entropy and the holographic principle in context of the AdS/CFT correspondence.
Quantum many-body problem with exponentially large degrees of freedom can be reduced to a tractable computational form by neural network method [G. Carleo and M. Troyer, Science 355 (2017) 602, arXiv:1606.02318.] The power of deep neural network (DNN) based on deep learning is clarified by mapping it to renormalization group (RG), which may shed lights on holographic principle by identifying a sequence of RG transformations to the AdS geometry. In this paper, we show that any network which reflects RG process has intrinsic hyperbolic geometry, and discuss the structure of entanglement encoded in the graph of DNN. We find the entanglement structure of DNN is of Ryu–Takayanagi form. Based on these facts, we argue that the emergence of holographic gravitational theory is related to deep learning process of the quantum-field theory.
For a given entanglement entropy of quantum field theory, we investigate how to reconstruct its dual geometry by applying the Ryu-Takayanagi formula and the deep learning method. In the holographic setup, the radial coordinate is identified with the energy scale of the dual quantum field theory. Therefore, the holographic dual geometry can describe how physical properties of a quantum field theory change along the renormalization group flow. Intriguingly, we show that the reconstructed geometry only from the entanglement entropy data can give us more information about other physical properties like thermodynamic quantities in the infrared region.
Deep learning is a broad set of techniques that uses multiple layers of representation to automatically learn relevant features directly from structured data. Recently, such techniques have yielded record-breaking results on a diverse set of difficult machine learning tasks in computer vision, speech recognition, and natural language processing. Despite the enormous success of deep learning, relatively little is understood theoretically about why these techniques are so successful at feature learning and compression. Here, we show that deep learning is intimately related to one of the most important and successful techniques in theoretical physics, the renormalization group (RG). RG is an iterative coarse-graining scheme that allows for the extraction of relevant features (i.e. operators) as a physical system is examined at different length scales. We construct an exact mapping from the variational renormalization group, first introduced by Kadanoff, and deep learning architectures based on Restricted Boltzmann Machines (RBMs). We illustrate these ideas using the nearest-neighbor Ising Model in one and two-dimensions. Our results suggests that deep learning algorithms may be employing a generalized RG-like scheme to learn relevant features from data.
The key idea behind the renormalization group (RG) transformation is that properties of physical systems with very different microscopic makeups can be characterized by a few universal parameters.However, finding a systematic way to construct RG transformation for particular systems remains difficult due to the many possible choices of the weight factors in the RG procedure.Here we show, by identifying the conditional distribution in the restricted Boltzmann machine and the weight factor distribution in the RG procedure, that a valid real-space RG transformation can be learned without prior knowledge of the physical system.This neural Monte Carlo RG algorithm allows for direct computation of the RG flow and critical exponents.Our results establish a solid connection between the RG transformation in physics and the deep architecture in machine learning, paving the way for further interdisciplinary research.
The renormalization group (RG) is an essential technique in statistical physics and quantum field theory, which considers scale-invariant properties of physical theories and how these theories' parameters change with scaling. Deep learning is a powerful computational technique that uses multi-layered neural networks to solve a myriad of complicated problems. Previous research suggests the possibility that unsupervised deep learning may be a form of RG flow, by being a layer-by-layer coarse graining of the original data. We examined this connection on a more rigorous basis for the simple example of Kadanoff block renormalization of the 2D nearest-neighbor Ising model, with our deep learning accomplished via Restricted Boltzmann Machines (RBMs). We developed extensive renormalization techniques for the 1D and 2D Ising model to provide a baseline for comparison. For the 1D Ising model, we successfully used Adam optimization on a correlation length loss function to learn the group flow, yielding results consistent with the analytical model for infinite N. For the 2D Ising model, we successfully generated Ising model samples using the Wolff algorithm, and performed the group flow using a quasi-deterministic method, validating these results by calculating the critical exponent ν. We then examined RBM learning of the Ising model layer by layer, finding a blocking structure in the learning that is qualitatively similar to RG. Lastly, we directly compared the weights of each layer from the learning to Ising spin renormalization, but found quantitative inconsistencies for the simple case of nearest-neighbor Ising models.
Renormalization group (RG) methods, which model the way in which the effective behavior of a system depends on the scale at which it is observed, are key to modern condensed-matter theory and particle physics. We compare the ideas behind the RG on the one hand and deep machine learning on the other, where depth and scale play a similar role. In order to illustrate this connection, we review a recent numerical method based on the RG---the multiscale entanglement renormalization ansatz (MERA)---and show how it can be converted into a learning algorithm based on a generative hierarchical Bayesian network model. Under the assumption---common in physics---that the distribution to be learned is fully characterized by local correlations, this algorithm involves only explicit evaluation of probabilities, hence doing away with sampling.