# Exploring the Energy Landscape of RBMs: Reciprocal Space Insights into Bosons, Hierarchical Learning and Symmetry Breaking

arXiv: https://arxiv.org/abs/2503.21536 · HTML: https://mlsanity.com/paper/2503.21536/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 22 of its 46 references that are in the map.
- First posted on arXiv 27 Mar 2025.

## Builds on (most cited first, 22 of 22)

- [Deep Unsupervised Learning using Nonequilibrium Thermodynamics](https://mlsanity.com/paper/1503.03585/) (2015)
- [Consistency Models](https://mlsanity.com/paper/2303.01469/) (2023)
- [Similarity of Neural Network Representations Revisited](https://mlsanity.com/paper/1905.00414/) (2019)
- [A Phase Transition in Diffusion Models Reveals the Hierarchical Nature of Data](https://mlsanity.com/paper/2402.16991/) (2024)
- [Do Convolutional Neural Networks Learn Class Hierarchy?](https://mlsanity.com/paper/1710.06501/) (2017)
- [Score-based Generative Models for Calorimeter Shower Simulation](https://mlsanity.com/paper/2206.11898/) (2022)
- [Discovering Phase Transitions with Unsupervised Learning](https://mlsanity.com/paper/1606.00318/) (2016)
- [The statistical thermodynamics of generative diffusion models: Phase transitions, symmetry breaking and critical instability](https://mlsanity.com/paper/2310.17467/) (2023)
- [Restricted Boltzmann Machine, recent advances and mean-field theory](https://mlsanity.com/paper/2011.11307/) (2020)
- [Equilibrium and non-Equilibrium regimes in the learning of Restricted Boltzmann Machines](https://mlsanity.com/paper/2105.13889/) (2021)
- [Thermodynamics of Restricted Boltzmann Machines and related learning dynamics](https://mlsanity.com/paper/1803.01960/) (2018)
- [The Landscape of Unfolding with Machine Learning](https://mlsanity.com/paper/2404.18807/) (2024)
- [Accurate and Conservative Estimates of MRF Log-likelihood using Reverse Annealing](https://mlsanity.com/paper/1412.8566/) (2014)
- [Learning Thermodynamics with Boltzmann Machines](https://mlsanity.com/paper/1606.02718/) (2016)
- [Cascade of phase transitions in the training of Energy-based models](https://mlsanity.com/paper/2405.14689/) (2024)
- [Disentangling representations in Restricted Boltzmann Machines without adversaries](https://mlsanity.com/paper/2206.11600/) (2022)
- [Statistical physics of unsupervised learning with prior knowledge in neural networks](https://mlsanity.com/paper/1911.02344/) (2019)
- [Nonequilbrium physics of generative diffusion models](https://mlsanity.com/paper/2405.11932/) (2024)
- [Conditioned quantum-assisted deep generative surrogate for particle-calorimeter interactions](https://mlsanity.com/paper/2410.22870/) (2024)
- [Dyson Brownian motion and random matrix dynamics of weight matrices during learning](https://mlsanity.com/paper/2411.13512/) (2024)
- [Using Deep LSD to build operators in GANs latent space with meaning in real space](https://mlsanity.com/paper/2102.05132/) (2021)
- [Generalized Lanczos method for systematic optimization of neural-network quantum states](https://mlsanity.com/paper/2502.01264/) (2025)

## Abstract (arXiv)

Deep generative models have become ubiquitous due to their ability to learn and sample from complex distributions. Despite the proliferation of various frameworks, the relationships among these models remain largely unexplored, a gap that hinders the development of a unified theory of AI learning. We address two central challenges: clarifying the connections between different deep generative models and deepening our understanding of their learning mechanisms. We focus on Restricted Boltzmann Machines (RBMs), known for their universal approximation capabilities for discrete distributions. By introducing a reciprocal space formulation, we reveal a connection between RBMs, diffusion processes, and coupled Bosons. We show that at initialization, the RBM operates at a saddle point, where the local curvature is determined by the singular values, whose distribution follows the Marcenko-Pastur law and exhibits rotational symmetry. During training, this rotational symmetry is broken due to hierarchical learning, where different degrees of freedom progressively capture features at multiple levels of abstraction. This leads to a symmetry breaking in the energy landscape, reminiscent of Landau theory. This symmetry breaking in the energy landscape is characterized by the singular values and the weight matrix eigenvector matrix. We derive the corresponding free energy in a mean-field approximation. We show that in the limit of infinite size RBM, the reciprocal variables are Gaussian distributed. Our findings indicate that in this regime, there will be some modes for which the diffusion process will not converge to the Boltzmann distribution. To illustrate our results, we trained replicas of RBMs with different hidden layer sizes using the MNIST dataset. Our findings bridge the gap between disparate generative frameworks and also shed light on the processes underpinning learning in generative models.