Relevance in the Renormalization Group and in Information Theory (2012.01447v1)

Abstract: 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 information-theoretic notion of relevance defined in the Information Bottleneck (IB) formalism of compression theory, and the field-theoretic relevance of the Renormalization Group. 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.

• The paper establishes a fundamental equivalence between IB relevance and RG scaling dimensions in statistical field theories.
• The authors employ IB and transfer matrix techniques to extract RG-relevant operators, demonstrating the approach on the 2D Ising model.
• The findings offer practical insights for machine learning applications in physics and deepen our understanding of information loss in field theories.

Relevance in the Renormalization Group and in Information Theory: An Expert Analysis

This paper investigates the intriguing relationship between the notion of relevance in information theory, particularly within the Information Bottleneck (IB) formalism, and the concept of relevance in the field-theoretic framework of the Renormalization Group (RG). The authors establish a profound equivalence between these two notions, aligning the information-theoretic relevance with the RG relevance in the context of statistical physical systems described by a field theory. This connection not only provides insights into the interpretation of ML methods applied to physical systems but also bridges a conceptual gap between distinct scientific disciplines.

The Theoretical Framework

The core contribution of the paper is the analytical demonstration that, for statistical physical systems governed by field theory, the relevant degrees of freedom identified using IB actually correspond to operators with the lowest scaling dimensions according to RG. This equivalence is significant given the historical separation of information theory, which traditionally focuses on efficient transmission of information agnostic to content, and RG, a cornerstone of theoretical physics that provides a systematic way to paper changes in physical systems as a function of scale.

To achieve this connection, the authors leverage the IB framework, which defines relevant information as the portion of a dataset that maintains correlations with an auxiliary relevance variable. By comparing this with RG's definition of relevance, which is determined by the scaling behavior of operators, the paper succeeds in unifying these theories through a shared focus on identifying "relevant" features.

Numerical and Analytical Validation

The theoretical predictions are validated both numerically and analytically. The authors apply their framework to classical systems such as the 2D Ising model, demonstrating that the IB solutions indeed extract the most RG-relevant operators. This is accomplished by establishing a mathematical correspondence using transfer matrix (TM) techniques, which relate the eigenvalues and eigenvectors of the TM to the scaling dimensions of primary operators in a conformal field theory (CFT) describing the system.

The numerical results obtained for the critical Ising model confirm the analytical prediction for the IB phase transition, providing strong evidence for the proposed equivalence. Furthermore, the detailed exploration of the IB equations reveals how different symmetries of the data, both explicit and emergent, manifest in the structure of the solutions.

Implications and Future Prospects

The implications of this paper are multifaceted. Practically, the findings offer a methodology for enhancing the interpretability of ML applications in physics by ensuring that the extracted features have a well-defined physical significance. This opens a pathway to potentially automate tasks such as deriving effective field theories and detecting hidden symmetries.

Theoretically, the paper advances our understanding of information loss in RG processes, linking it to a formalism that is quantitative and constructive. This connection invites further exploration in broader contexts, including disordered and non-equilibrium systems, where traditional RG methods face challenges.

Moving forward, the integration of information-theoretical concepts with computationally efficient ML techniques presents an exciting frontier for both physics and computer science. The potential to apply such hybrid approaches to experimental data and complex systems remains a promising avenue for further research, aiming to harness the strengths of both theoretical analysis and ML-driven data processing.

In summary, the paper presents a well-founded and comprehensively validated bridge between RG relevance and IB within information theory, expanding the toolbox available to researchers in both fields and setting the stage for future interdisciplinary advancements.