[Paper Review] Information geometric complexity of entropic motion on curved statistical manifolds

Carlo Cafaro|arXiv (Cornell University)|Aug 21, 2013

Statistical Mechanics and Entropy50 references3 citations

TL;DR

This paper introduces an information geometric complexity measure for geodesic paths on curved statistical manifolds in the context of entropic dynamics, using Riemannian geometry to quantify the complexity of probabilistic models. The key contribution is a framework that links thermodynamic depth and logical depth to information-geometric structure, showing that complexity arises from the difficulty of path reconstruction in statistical manifolds, with implications for model selection and complexity softening in quantum embeddings.

ABSTRACT

Physical systems behave according to their underlying dynamical equations which, in turn, can be identified from experimental data. Explaining data requires selecting mathematical models that best capture the data regularities. Identifying dynamical equations from the available data and statistical model selection are both very difficult tasks. Motivated by these fundamental links among physical systems, dynamical equations, experimental data and statistical modeling, we discuss in this invited Contribution our information geometric measure of complexity of geodesic paths on curved statistical manifolds underlying the entropic dynamics of classical physical systems described by probability distributions. We also provide several illustrative examples of entropic dynamical models used to infer macroscopic predictions when only partial knowledge of the microscopic nature of the system is available. Finally, we present entropic arguments to briefly address complexity softening effects due to statistical embedding procedures.

Motivation & Objective

• To develop a geometric measure of complexity for geodesic paths in entropic dynamics on curved statistical manifolds.
• To address the challenge of inferring dynamical equations from incomplete data using information-geometric principles.
• To explore how statistical embedding procedures may lead to complexity softening effects.
• To establish a bridge between thermodynamics, information geometry, and complexity in physical systems.

Proposed method

• The paper employs the information-geometric approach to classical (IGAC) dynamics, modeling systems via probability distributions on Riemannian statistical manifolds.
• It defines complexity as the geometric length of geodesic paths in the manifold, reflecting the difficulty of reconstructing a system's evolution from data.
• The framework uses Fisher information metric to define the Riemannian structure of the statistical manifold, enabling curvature-based analysis of dynamical complexity.
• It applies entropic arguments to analyze how embedding classical models into higher-dimensional or quantum-like structures can reduce effective complexity.
• The method draws analogies between thermodynamic depth and logical depth, embedding them within information-geometric flows on manifolds.
• It considers the role of quantum features—such as Heisenberg’s uncertainty principle and quantum state discrimination—in enabling complexity softening without violating the second law of thermodynamics.

Experimental results

Research questions

• RQ1 How can the complexity of geodesic paths in entropic dynamics be quantified using information geometry on curved statistical manifolds?
• RQ2 What is the relationship between thermodynamic depth, logical depth, and information-geometric complexity in probabilistic models?
• RQ3 In what way do statistical embedding procedures lead to complexity softening, and how can this be explained via entropic arguments?
• RQ4 Can quantum features such as non-orthogonal state discrimination and uncertainty principles be linked to complexity reduction in a thermodynamically consistent way?
• RQ5 How might thermodynamics serve as a unifying framework for different complexity measures in classical and quantum systems?

Key findings

• The information-geometric complexity measure quantifies the difficulty of reconstructing a system’s dynamical evolution through the geometric length of geodesic paths on statistical manifolds.
• The framework shows that complexity is not monotonic with randomness or order, but arises from structured, non-trivial paths in the manifold, aligning with the idea that complex systems require difficult reconstruction paths.
• Statistical embedding procedures can reduce effective complexity, a phenomenon termed 'complexity softening,' which is entropically explainable and consistent with thermodynamic constraints.
• Quantum features such as Heisenberg’s uncertainty principle and the possibility of distinguishing nonorthogonal states are shown to potentially enable complexity softening, provided they do not violate the second law of thermodynamics.
• The paper establishes a conceptual and formal link between thermodynamics, information geometry, and complexity, suggesting that thermodynamic consistency is a key internal consistency check for any complexity measure.
• The results suggest that entropic dynamics on curved manifolds naturally incorporate principles of irreversibility and memory cost, aligning with thermodynamic depth and logical depth as complexity indicators.