Editorial of the Research Topic "Advances in Information Geometry: Beyond the Conventional Approach"

Florio M. Ciaglia^1*Fabio Di Cosmo^2,3Alberto Ibort^1,3Jun Suzuki⁴

• ¹Universidad Carlos III de Madrid Departamento de Matematicas, Leganés, Spain
• ²Universidad de Alcala, Alcala de Henares, Spain
• ³Instituto de Ciencias Matematicas, Madrid, Spain
• ⁴Denki Tsushin Daigaku, Chofu, Japan

Information Geometry (IG) is an active interdisciplinary field, employing the tools of differential geometry to explore the structure of classical and quantum statistical models of probability distributions and quantum states, respectively. It has provided powerful insights into the geometrical aspects of classical and quantum statistical inference, machine learning, signal processing, and neural networks, primarily by endowing statistical models with geometric structures like Riemannian metrics (e.g., Fisher-Rao, Bures-Helstrom metric) and affine connections (e.g., Amari-Chentsov α-connections). This geometric perspective allows for a deeper understanding of parameter estimation, model selection, and learning dynamics.The Research Topic Advances in Information Geometry: Beyond the Conventional Approach was conceived to capture the ever-evolving spirit of innovation of the investigation in IG. It aims to showcase some of the most innovative research that pushes the boundaries of traditional IG, whether through novel theoretical extensions, the exploration of new application domains, or the development of methodologies that challenge standard assumptions. The collection of articles within this Research Topic offers a glimpse into these exciting advancements, each contributing a unique perspective on how IG can address contemporary challenges and open new avenues of investigation moving "beyond the conventional approach."The work [1] presents a novel link between information geometry-related concepts, thermodynamics, and the rapidly advancing field of deep learning. The authors propose a thermodynamic analogy to analyze the dynamics of parameters in Convolutional Neural Networks (CNNs) that allows for a rigorous definition of "temperature" for convolutional filters. Their research demonstrates that high-temperature filters have a minimal impact on model performance when removed, whereas removing low-temperature filters significantly affects accuracy and loss decay. This insight leads to a practical, temperature-based filter pruning technique. This application of thermodynamic and statistical mechanical thinking, which shares deep conceptual roots with information geometry (e.g., geometrization of state spaces), to the architecture and optimization of deep learning models is a significant step beyond conventional applications of IG, showcasing its potential utility in understanding and refining complex neural networks.The article [2] investigates the foundation quantum IG. The conventional approach often relies on quantum statistical models for finite-dimensional quantum systems. The author, however, considers manifolds of faithful normal states on a sigma-finite von Neumann algebra in standard form. The paper generalizes the concept of an exponential arc connecting quantum states, using a given relative entropy (divergence function).A key finding is the uniqueness (up to an additive constant) of the generator of such an arc. Specifically, when using Araki's relative entropy, any self-adjoint element of the von Neumann algebra can generate an exponential arc. This formulation demonstrates that the metric derived from Araki's relative entropy aligns with the Kubo-Mori metric, crucial in linear response theory. These submanifolds, formed by states connected via these exponential arcs, represent a quantum generalization of dually flat statistical manifolds. This contribution outlines a promising avenue of development of IG to more general, potentially infinite-dimensional, quantum systems, moving beyond the standard density matrix formalism.The article [3] discusses the foundational aspects of Bayesian statistics concerning noninformative priors. While the Jeffreys prior is a cornerstone in this area, derived from the Fisher information metric, the author explores an interesting alternative: a noninformative prior based on the χ 2divergence. This prior, an extension of Bernardo's reference prior, offers a different geometric interpretation.The paper elucidates that this prior corresponds to a parallel volume element within the framework of information geometry. Furthermore, in the context of flat model manifolds, it can be expressed as a power of the Jeffreys prior. This work moves beyond conventional reliance on the Jeffreys prior by investigating the geometric underpinnings and properties of priors derived from alternative divergence measures, thus broadening the toolkit for objective Bayesian inference. This survey, however, explores the information geometry of Markov kernels, which represent conditional probabilities or stochastic transitions, thereby extending IG to dynamical processes. The authors present a self-contained treatment of the foundational concepts, including information projections and Nagaoka's construction of Fisher metrics and dual affine connections on sets of irreducible stochastic matrices.The survey also discusses recent advancements, such as geometric structures arising from time reversibility, lumpability of Markov chains, and tree models. Applications in parameter estimation, hypothesis testing, large deviation theory, and the maximum entropy principle are highlighted. This work clearly highlights how information geometry can be extended beyond static distributions to characterize the structure of transformations and evolving systems.The journey "beyond the conventional approach" of Information Geometry is an ongoing one. The articles collected in this Research Topic, while diverse in their specific focus, collectively highlights the broad scope of Information Geometry. They illustrate that the field is not static, but is actively incorporating new mathematical tools, addressing more complex systems, and finding relevance in an increasingly diverse range of scientific and technological disciplines. The insights presented here are expected to push further research, inspiring new theoretical developments and practical applications that continue to redefine the field of Information Geometry.