Data-Driven Energy-Based Learning via Gibbs Measures on Hierarchical Structures

The authors propose a departure from standard empirical risk minimization, which typically seeks a single optimal parameter. Instead, they treat the empirical loss function as an interaction potential in an energy-based model defined on a Cayley tree. By mapping the learning problem to the statistical mechanics of Gibbs measures, the dataset is viewed as generating a probabilistic landscape of possible inference states rather than a single point estimate.

To characterize these learning states, the authors derive nonlinear integral fixed-point equations that ensure the consistency of finite-volume Gibbs distributions. For translation-invariant solutions, the problem is reduced to analyzing positive compact operators induced by data-dependent kernels. This allows the authors to apply tools from spectral theory, such as the Krein-Rutman theorem, to establish conditions for the existence and uniqueness of equilibrium learning regimes. The authors also perform numerical experiments to observe how non-separable kernels lead to multiple solution branches.

The study demonstrates that hierarchical learning systems can exhibit phase-transition phenomena. Specifically, beyond a critical inverse temperature, the system may transition from a unique equilibrium state to a regime where multiple Gibbs measures coexist, representing distinct, competing prediction regimes. The authors show that the number of these measures acts as an order parameter for the learning system, providing a rigorous mathematical basis for understanding multi-modal learning and latent state separation in hierarchical models.

This work bridges the gap between empirical machine learning and the statistical mechanics of interacting systems. By reinterpreting learning as the study of equilibrium states on hierarchical structures, the framework provides a new lens through which to view uncertainty and multi-modality in complex models. It suggests that the emergence of multiple learning regimes is not merely a numerical artifact but a fundamental property of the probabilistic landscape defined by the data.