# Neural Sampling in Hierarchical Exponential-family Energy-based Models

arXiv: https://arxiv.org/abs/2310.08431 · HTML: https://mlsanity.com/paper/2310.08431/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 17 of its 64 references that are in the map.
- First posted on arXiv 12 Oct 2023.

## Builds on (most cited first, 17 of 17)

- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [Generative Adversarial Networks](https://mlsanity.com/paper/2203.00667/) (2022)
- [Fashion-MNIST: a Novel Image Dataset for Benchmarking Machine Learning Algorithms](https://mlsanity.com/paper/1708.07747/) (2017)
- [Improved Techniques for Training GANs](https://mlsanity.com/paper/1606.03498/) (2016)
- [Equilibrium Propagation: Bridging the Gap Between Energy-Based Models and Backpropagation](https://mlsanity.com/paper/1602.05179/) (2016)
- [Convergence of Langevin MCMC in KL-divergence](https://mlsanity.com/paper/1705.09048/) (2017)
- [Classification and Geometry of General Perceptual Manifolds](https://mlsanity.com/paper/1710.06487/) (2017)
- [Maximum Entropy Generators for Energy-Based Models](https://mlsanity.com/paper/1901.08508/) (2019)
- [The Neural Coding Framework for Learning Generative Models](https://mlsanity.com/paper/2012.03405/) (2020)
- [Associative Memories via Predictive Coding](https://mlsanity.com/paper/2109.08063/) (2021)
- [Deep Directed Generative Models with Energy-Based Probability Estimation](https://mlsanity.com/paper/1606.03439/) (2016)
- [Predictive Coding beyond Gaussian Distributions](https://mlsanity.com/paper/2211.03481/) (2022)
- [Breaking Reversibility Accelerates Langevin Dynamics for Global Non-Convex Optimization](https://mlsanity.com/paper/1812.07725/) (2018)
- [The implications of perception as probabilistic inference for correlated neural variability during behavior](https://mlsanity.com/paper/1409.0257/) (2014)
- [The Hamiltonian brain: efficient probabilistic inference with excitatory-inhibitory neural circuit dynamics](https://mlsanity.com/paper/1407.0973/) (2014)
- [A probabilistic population code based on neural samples](https://mlsanity.com/paper/1811.09739/) (2018)

## Built on by (most cited first, 4 of 4)

- [Divide-and-Conquer Predictive Coding: a structured Bayesian inference algorithm](https://mlsanity.com/paper/2408.05834/) (2024)
- [Neural Langevin Machine: a local asymmetric learning rule can be creative](https://mlsanity.com/paper/2506.23546/) (2025)
- [Predictive Learning in Energy-based Models with Attractor Structures](https://mlsanity.com/paper/2501.13997/) (2025)
- [Leakage and Second-Order Dynamics Improve Hippocampal RNN Replay](https://mlsanity.com/paper/2602.18401/) (2026)

## Abstract (arXiv)

Bayesian brain theory suggests that the brain employs generative models to understand the external world. The sampling-based perspective posits that the brain infers the posterior distribution through samples of stochastic neuronal responses. Additionally, the brain continually updates its generative model to approach the true distribution of the external world. In this study, we introduce the Hierarchical Exponential-family Energy-based (HEE) model, which captures the dynamics of inference and learning. In the HEE model, we decompose the partition function into individual layers and leverage a group of neurons with shorter time constants to sample the gradient of the decomposed normalization term. This allows our model to estimate the partition function and perform inference simultaneously, circumventing the negative phase encountered in conventional energy-based models (EBMs). As a result, the learning process is localized both in time and space, and the model is easy to converge. To match the brain's rapid computation, we demonstrate that neural adaptation can serve as a momentum term, significantly accelerating the inference process. On natural image datasets, our model exhibits representations akin to those observed in the biological visual system. Furthermore, for the machine learning community, our model can generate observations through joint or marginal generation. We show that marginal generation outperforms joint generation and achieves performance on par with other EBMs.