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Bulletin of the American Physical Society

Real-space mutual information neural estimation algorithm for single-step extraction of renormalisation group-relevant degrees of freedom

Doruk Efe Gökmen, Zohar Ringel, Sebastian D. Huber, Maciej Koch-Janusz

Published Mar 15, 2021• 0 citations
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Mutual information, neural networks and the renormalization group

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Symmetries and phase diagrams with real-space mutual information neural estimation

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Real-space mutual information (RSMI) was shown to be an important quantity, formally and from a numerical standpoint, in finding coarse-grained descriptions of physical systems. It very generally quantifies spatial correlations and can give rise to constructive algorithms extracting relevant degrees of freedom. Efficient and reliable estimation or maximization of RSMI is, however, numerically challenging. A recent breakthrough in theoretical machine learning has been the introduction of variational lower bounds for mutual information, parametrized by neural networks. Here we describe in detail how these results can be combined with differentiable coarse-graining operations to develop a single unsupervised neural-network-based algorithm, the RSMI-NE, efficiently extracting the relevant degrees of freedom in the form of the operators of effective field theories, directly from real-space configurations. We study the information contained in the statistical ensemble of constructed coarse-graining transformations and its recovery from partial input data using a secondary machine learning analysis applied to this ensemble. In particular, we show how symmetries, also emergent, can be identified. We demonstrate the extraction of the phase diagram and the order parameters for equilibrium systems and consider also an example of a nonequilibrium problem.

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Real-space mutual information (RSMI) has been shown to be an important quantity, both formally and from numerical standpoint, in constructing coarse-grained descriptions of physical systems. It very generally quantifies spatial correlations, and can give rise to constructive algorithms extracting relevant degrees of freedom. Efficient and reliable estimation or maximization of RSMI is, however, numerically challenging. A recent breakthrough in theoretical machine learning has been the introduction of variational lower bounds for mutual information, parametrized by neural networks. Here we describe in detail how these results can be combined with differentiable coarse-graining operations to develop a single unsupervised neural-network based algorithm, the RSMI-NE, efficiently extracting the relevant degrees of freedom in the form of the operators of effective field theories, directly from real-space configurations. We study the information, e.g. about the symmetries, contained in the ensemble of constructed coarse-graining transformations, and its practical recovery from partial input data using a secondary machine learning analysis applied to this ensemble. We demonstrate how the phase diagram and the order parameters for equilibrium systems are extracted, and consider also an example of a non-equilibrium problem.

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Recently, a novel real-space renormalization group (RG) algorithm was introduced.By maximizing an information-theoretic quantity, the real-space mutual information, the algorithm identifies the relevant lowenergy degrees of freedom.Motivated by this insight, we investigate the information-theoretic properties of coarse-graining procedures for both translationally invariant and disordered systems.We prove that a perfect real-space mutual information coarse graining does not increase the range of interactions in the renormalized Hamiltonian, and, for disordered systems, it suppresses the generation of correlations in the renormalized disorder distribution, being in this sense optimal.We empirically verify decay of those measures of complexity as a function of information retained by the RG, on the examples of arbitrary coarse grainings of the clean and random Ising chain.The results establish a direct and quantifiable connection between properties of RG viewed as a compression scheme and those of physical objects, i.e., Hamiltonians and disorder distributions.We also study the effect of constraints on the number and type of coarse-grained degrees of freedom on a generic RG procedure.

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The Density Matrix Renormalization Group Algorithm for Strongly Correlated Systems: A Generic Implementation

Gonzalo A. Álvarez

Bulletin of the American Physical Society•Mar 17, 2009•0 citations
Real-space mutual information (RSMI) was shown to be an important quantity, formally and from a numerical standpoint, in finding coarse-grained descriptions of physical systems. It very generally quantifies spatial correlations and can give rise to constructive algorithms extracting relevant degrees of freedom. Efficient and reliable estimation or maximization of RSMI is, however, numerically challenging. A recent breakthrough in theoretical machine learning has been the introduction of variational lower bounds for mutual information, parametrized by neural networks. Here we describe in detail how these results can be combined with differentiable coarse-graining operations to develop a single unsupervised neural-network-based algorithm, the RSMI-NE, efficiently extracting the relevant degrees of freedom in the form of the operators of effective field theories, directly from real-space configurations. We study the information contained in the statistical ensemble of constructed coarse-graining transformations and its recovery from partial input data using a secondary machine learning analysis applied to this ensemble. In particular, we show how symmetries, also emergent, can be identified. We demonstrate the extraction of the phase diagram and the order parameters for equilibrium systems and consider also an example of a nonequilibrium problem.
Real-space mutual information (RSMI) has been shown to be an important quantity, both formally and from numerical standpoint, in constructing coarse-grained descriptions of physical systems. It very generally quantifies spatial correlations, and can give rise to constructive algorithms extracting relevant degrees of freedom. Efficient and reliable estimation or maximization of RSMI is, however, numerically challenging. A recent breakthrough in theoretical machine learning has been the introduction of variational lower bounds for mutual information, parametrized by neural networks. Here we describe in detail how these results can be combined with differentiable coarse-graining operations to develop a single unsupervised neural-network based algorithm, the RSMI-NE, efficiently extracting the relevant degrees of freedom in the form of the operators of effective field theories, directly from real-space configurations. We study the information, e.g. about the symmetries, contained in the ensemble of constructed coarse-graining transformations, and its practical recovery from partial input data using a secondary machine learning analysis applied to this ensemble. We demonstrate how the phase diagram and the order parameters for equilibrium systems are extracted, and consider also an example of a non-equilibrium problem.
Recently, a novel real-space renormalization group (RG) algorithm was introduced.By maximizing an information-theoretic quantity, the real-space mutual information, the algorithm identifies the relevant lowenergy degrees of freedom.Motivated by this insight, we investigate the information-theoretic properties of coarse-graining procedures for both translationally invariant and disordered systems.We prove that a perfect real-space mutual information coarse graining does not increase the range of interactions in the renormalized Hamiltonian, and, for disordered systems, it suppresses the generation of correlations in the renormalized disorder distribution, being in this sense optimal.We empirically verify decay of those measures of complexity as a function of information retained by the RG, on the examples of arbitrary coarse grainings of the clean and random Ising chain.The results establish a direct and quantifiable connection between properties of RG viewed as a compression scheme and those of physical objects, i.e., Hamiltonians and disorder distributions.We also study the effect of constraints on the number and type of coarse-grained degrees of freedom on a generic RG procedure.