Spherical Boltzmann machines: a solvable theory of learning and generation in energy-based models

• URL: http://arxiv.org/abs/2605.09031v1
• Date: Sat, 09 May 2026 16:15:18 GMT
• Title: Spherical Boltzmann machines: a solvable theory of learning and generation in energy-based models
• Abstract summary: Energy-based models (EBMs) are flexible generative architectures inspired by statistical physics.<n>We analyze a solvable EBM in the high-dimensional limit: the spherical Boltzmann machine (SBM)
• Score: 3.661853317875675
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Energy-based models (EBMs) are flexible generative architectures inspired by statistical physics, but their learning and generative properties remain poorly understood. Here, we analyze a solvable EBM in the high-dimensional limit: the spherical Boltzmann machine (SBM). Combining tools from random matrix theory and dynamical mean-field theory, we: solve exact equations describing the training dynamics of the SBM; compute the Bayesian evidence, which acts as a partition function in parameter space and encodes global properties of the trained model; and uncover cascades of phase transitions that occur both during training and as a function of hyperparameters, related to successive alignment and condensation of the top modes of the coupling matrix to the data. We connect these transitions to sampling-time generative phenomena in a teacher-student scenario, including: sampling temperature tuning, double descent as a function of regularization strength, tempered posterior effects, and out-of-equilibrium effects during training that induce biases in the trained model. We provide numerical evidence demonstrating that all these phenomena appear in standard generative architectures, beyond the SBM.

Related papers

• EFF-Grasp: Energy-Field Flow Matching for Physics-Aware Dexterous Grasp Generation [54.09734511705173]
  EFF-Grasp is a Flow-Matching synthesis-based framework for physics-aware dexterous grasp generation.<n>We reformulate grasp as a deterministic ordinary differential equation (ODE) process, which enables efficient and stable generation through smooth probability flows.<n> EFF-Grasp achieves superior performance in grasp quality and physical feasibility, while requiring substantially fewer sampling steps than diffusion-based baselines.
  arXiv  Detail & Related papers  (2026-03-17T06:11:56Z)
• Introduction to Sachdev-Ye-Kitaev Model: A Strongly Correlated System Perspective [0.0]
  The Sachdev-Ye-Kitaev (SYK) model provides an analytically tractable framework for exotic strongly correlated phases.<n>This review offers a pedagogical introduction to the SYK physics, highlighting its unique capacity to model textitstrange metals<n>We transform Majorana and complex fermion variants into training grounds for modern many-body physics techniques.
  arXiv  Detail & Related papers  (2025-07-09T18:10:54Z)
• Models of Heavy-Tailed Mechanistic Universality [62.107333654304014]
  We propose a family of random matrix models to explore attributes that give rise to heavy-tailed behavior in trained neural networks.<n>Under this model, spectral densities with power laws on tails arise through a combination of three independent factors.<n> Implications of our model on other appearances of heavy tails, including neural scaling laws, trajectories, and the five-plus-one phases of neural network training, are discussed.
  arXiv  Detail & Related papers  (2025-06-04T00:55:01Z)
• Energy Matching: Unifying Flow Matching and Energy-Based Models for Generative Modeling [5.787270247104665]
  Current state-of-the-art generative models map noise to data distributions by matching flows or scores.<n>Here, we propose Energy Matching, a framework that endows flow-based approaches with the flexibility of EBMs.
  arXiv  Detail & Related papers  (2025-04-14T18:10:58Z)
• Parametric model reduction of mean-field and stochastic systems via higher-order action matching [1.1509084774278489]
  We learn models of population dynamics of physical systems that feature gradient and mean-field effects. We show that our approach accurately predicts population dynamics over a wide range of parameters and outperforms state-of-the-art diffusion-based and flow-based modeling.
  arXiv  Detail & Related papers  (2024-10-15T19:05:28Z)
• Latent Space Energy-based Neural ODEs [73.01344439786524]
  This paper introduces novel deep dynamical models designed to represent continuous-time sequences.<n>We train the model using maximum likelihood estimation with Markov chain Monte Carlo.<n> Experimental results on oscillating systems, videos and real-world state sequences (MuJoCo) demonstrate that our model with the learnable energy-based prior outperforms existing counterparts.
  arXiv  Detail & Related papers  (2024-09-05T18:14:22Z)
• Event-Based Simulation of Stochastic Memristive Devices for Neuromorphic Computing [41.66366715982197]
  We build a general model of memristors suitable for the simulation of event-based systems. We extend an existing general model of memristors to an event-driven setting. We demonstrate an approach for fitting the parameters of the event-based model to the drift model.
  arXiv  Detail & Related papers  (2024-06-14T13:17:19Z)
• Capturing dynamical correlations using implicit neural representations [85.66456606776552]
  We develop an artificial intelligence framework which combines a neural network trained to mimic simulated data from a model Hamiltonian with automatic differentiation to recover unknown parameters from experimental data. In doing so, we illustrate the ability to build and train a differentiable model only once, which then can be applied in real-time to multi-dimensional scattering data.
  arXiv  Detail & Related papers  (2023-04-08T07:55:36Z)
• Physics-Integrated Variational Autoencoders for Robust and Interpretable Generative Modeling [86.9726984929758]
  We focus on the integration of incomplete physics models into deep generative models. We propose a VAE architecture in which a part of the latent space is grounded by physics. We demonstrate generative performance improvements over a set of synthetic and real-world datasets.
  arXiv  Detail & Related papers  (2021-02-25T20:28:52Z)
• A physics-informed operator regression framework for extracting data-driven continuum models [0.0]
  We present a framework for discovering continuum models from high fidelity molecular simulation data. Our approach applies a neural network parameterization of governing physics in modal space. We demonstrate the effectiveness of our framework for a variety of physics, including local and nonlocal diffusion processes and single and multiphase flows.
  arXiv  Detail & Related papers  (2020-09-25T01:13:51Z)
• Training Deep Energy-Based Models with f-Divergence Minimization [113.97274898282343]
  Deep energy-based models (EBMs) are very flexible in distribution parametrization but computationally challenging. We propose a general variational framework termed f-EBM to train EBMs using any desired f-divergence. Experimental results demonstrate the superiority of f-EBM over contrastive divergence, as well as the benefits of training EBMs using f-divergences other than KL.
  arXiv  Detail & Related papers  (2020-03-06T23:11:13Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.