A Centre for Complexity Science Seminar.

Based on the recent works [1–3] and most recently [4] we introduce and explore a novel, first-principled formulation to understand physical systems that cannot be properly described by classic information-theoretic quantities such as Shannon’s entropy. Usage of information-theoretic methods to understand physical phenomena has continued to gain traction in the physics community, being promising for enabling new effective methods of analysis and opening new avenues of inquiry, from the discovery of the Ryu-Takayanagi formula for the entanglement entropy more than a decade ago to powerful tools to study quantum gravity and quantum field theory.

However, many physical systems of interest cannot be described by the standard tools of information theory, and therefore extensions are required. There have been developments in non-Shannonian methods, but the lack of a unifying principle or encompassing framework renders such approaches — despite their efficacy — heuristic at best.

In this talk, we address this knowledge gap by introducing a method grounded in the maximum entropy principle applied to curved statistical manifolds. One of the consequences of this extension is the induction of higher-order interactions.

This feature opens the door to a plethora of future theoretical and practical investigations ranging from neural-networks to neuron systems. Importantly, it allows for the relatively simple construction of a model that exhibits very complex phenomena, including the appearance of explosive phase transitions.

References:

[1] Morales, P.A. and Rosas, F.E., ”Generalization of the maximum entropy principle for curved statistical manifolds”. Physical Review Research, 3, 033216 (2021).

[2] Morales, P.A., Korbel, J., Rosas, F.E., “Geometric Structures Induced by Deformations of the Legendre Transform”, Entropy, 25(4), 678 (2023).

[3] Morales, P.A., Korbel, J., Rosas, F.E., “Thermodynamics of exponential Kolmogorov-Nagumo averages”, New J. Phys., 25, 073011 (2023).

[4] Aguilera, M., Morales, P.A., Shimazaki, H. and Rosas, F.E., “Inducing Higher-Order Interactions in Neural Networks via Curved Statistical Manifolds”. to be submitted.