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arXiv:2101.11633 · Physical Review Letters

Statistical Physics through the Lens of Real-Space Mutual Information

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

Published Dec 6, 2021• 21 citations
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Abstract

Identifying the relevant degrees of freedom in a complex physical system is a key stage in developing effective theories in and out of equilibrium. The celebrated renormalization group provides a framework for this, but its practical execution in unfamiliar systems is fraught with ad hoc choices, whereas machine learning approaches, though promising, lack formal interpretability. Here we present an algorithm employing state-of-the-art results in machine-learning-based estimation of information-theoretic quantities, overcoming these challenges, and use this advance to develop a new paradigm in identifying the most relevant operators describing properties of the system. We demonstrate this on an interacting model, where the emergent degrees of freedom are qualitatively different from the microscopic constituents. Our results push the boundary of formally interpretable applications of machine learning, conceptually paving the way toward automated theory building.

DOI
10.1103/physrevlett.127.240603

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Identifying the relevant degrees of freedom in a complex physical system is a key stage in developing effective theories in and out of equilibrium. The celebrated renormalization group provides a framework for this, but its practical execution in unfamiliar systems is fraught with ad hoc choices, whereas machine learning approaches, though promising, lack formal interpretability. Here we present an algorithm employing state-of-the-art results in machine-learning-based estimation of information-theoretic quantities, overcoming these challenges, and use this advance to develop a new paradigm in identifying the most relevant operators describing properties of the system. We demonstrate this on an interacting model, where the emergent degrees of freedom are qualitatively different from the microscopic constituents. Our results push the boundary of formally interpretable applications of machine learning, conceptually paving the way toward automated theory building.
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.
The analysis of complex physical systems hinges on the ability to extract the relevant degrees of freedom from among the many others. Though much hope is placed in machine learning, it also brings challenges, chief of which is interpretability. It is often unclear what relation, if any, the architecture- and training-dependent learned "relevant" features bear to standard objects of physical theory. Here we report on theoretical results which may help to systematically address this issue: we establish equivalence between the field-theoretic relevance of the renormalization group, and an information-theoretic notion of relevance we define using the information bottleneck (IB) formalism of compression theory. We show analytically that for statistical physical systems described by a field theory the relevant degrees of freedom found using IB compression indeed correspond to operators with the lowest scaling dimensions. We confirm our field theoretic predictions numerically. We study dependence of the IB solutions on the physical symmetries of the data. Our findings provide a dictionary connecting two distinct theoretical toolboxes, and an example of constructively incorporating physical interpretability in applications of deep learning in physics.
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.
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.
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In physics, one attempts to infer the rules governing a system given only the results of imperfect measurements. Hence, microscopic theories may be effectively indistinguishable experimentally. We develop an operationally motivated procedure to identify the corresponding equivalence classes of states, and argue that the renormalization group (RG) arises from the inherent ambiguities associated with the classes: one encounters flow parameters as, e.g., a regulator, a scale, or a measure of precision, which specify representatives in a given equivalence class. This provides a unifying framework and reveals the role played by information in renormalization. We validate this idea by showing that it justifies the use of low-momenta n -point functions as statistically relevant observables around a Gaussian hypothesis. These results enable the calculation of distinguishability in quantum field theory. Our methods also provide a way to extend renormalization techniques to effective models which are not based on the usual quantum-field formalism, and elucidates the relationships between various type of RG.