# Kolmogorov-Arnold Energy Models: Fast, Interpretable Generative Modeling

arXiv: https://arxiv.org/abs/2506.14167 · HTML: https://mlsanity.com/paper/2506.14167/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 27 of its 84 references that are in the map.
- First posted on arXiv 17 Jun 2025.

## Builds on (most cited first, 27 of 27)

- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [Very Deep Convolutional Networks for Large-Scale Image Recognition](https://mlsanity.com/paper/1409.1556/) (2014)
- [Generative Adversarial Networks](https://mlsanity.com/paper/2203.00667/) (2022)
- [Rethinking the Inception Architecture for Computer Vision](https://mlsanity.com/paper/1512.00567/) (2015)
- [Fashion-MNIST: a Novel Image Dataset for Benchmarking Machine Learning Algorithms](https://mlsanity.com/paper/1708.07747/) (2017)
- [Deep Learning Face Attributes in the Wild](https://mlsanity.com/paper/1411.7766/) (2014)
- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [Perceptual Losses for Real-Time Style Transfer and Super-Resolution](https://mlsanity.com/paper/1603.08155/) (2016)
- [Variational Inference with Normalizing Flows](https://mlsanity.com/paper/1505.05770/) (2015)
- [Normalizing Flows for Probabilistic Modeling and Inference](https://mlsanity.com/paper/1912.02762/) (2019)
- [Neural Spline Flows](https://mlsanity.com/paper/1906.04032/) (2019)
- [Semi-Supervised Learning with Deep Generative Models](https://mlsanity.com/paper/1406.5298/) (2014)
- [Mechanistic Interpretability for AI Safety -- A Review](https://mlsanity.com/paper/2404.14082/) (2024)
- [Cyclical Annealing Schedule: A Simple Approach to Mitigating KL Vanishing](https://mlsanity.com/paper/1903.10145/) (2019)
- [Effectively Unbiased FID and Inception Score and where to find them](https://mlsanity.com/paper/1911.07023/) (2019)
- [Learning Non-Convergent Non-Persistent Short-Run MCMC Toward Energy-Based Model](https://mlsanity.com/paper/1904.09770/) (2019)
- [Inference Suboptimality in Variational Autoencoders](https://mlsanity.com/paper/1801.03558/) (2018)
- [Kolmogorov-Arnold Networks are Radial Basis Function Networks](https://mlsanity.com/paper/2405.06721/) (2024)
- [On the Anatomy of MCMC-Based Maximum Likelihood Learning of Energy-Based Models](https://mlsanity.com/paper/1903.12370/) (2019)
- [Learning Latent Space Energy-Based Prior Model](https://mlsanity.com/paper/2006.08205/) (2020)
- [MLIR: A Compiler Infrastructure for the End of Moore's Law](https://mlsanity.com/paper/2002.11054/) (2020)
- [Instead of Rewriting Foreign Code for Machine Learning, Automatically Synthesize Fast Gradients](https://mlsanity.com/paper/2010.01709/) (2020)
- [Demystifying μ](https://mlsanity.com/paper/2401.01096/) (2024)
- [Learning Multimodal Latent Generative Models with Energy-Based Prior](https://mlsanity.com/paper/2409.19862/) (2024)
- [Learning Latent Space Hierarchical EBM Diffusion Models](https://mlsanity.com/paper/2405.13910/) (2024)
- [Persistently Trained, Diffusion-assisted Energy-based Models](https://mlsanity.com/paper/2304.10707/) (2023)
- [A structured proof of Kolmogorov's Superposition Theorem](https://mlsanity.com/paper/2105.00408/) (2021)

## Built on by (most cited first, 1 of 1)

- [Reasoning as Energy Minimization over Structured Latent Trajectories](https://mlsanity.com/paper/2603.28248/) (2026)

## Abstract (arXiv)

Generative models typically rely on either simple latent priors (e.g., Variational Autoencoders, VAEs), which are efficient but limited, or expressive iterative samplers (e.g., Diffusion and Energy-based Models), which are costly and opaque. We introduce a new unsupervised model, the Kolmogorov-Arnold Energy Model (KAEM), to bridge this trade-off and provide new opportunities for interpretability. Based on a novel adaptation of the Kolmogorov-Arnold Representation Theorem, KAEM imposes a univariate latent prior, enabling fast and exact inference via the inverse transform method. On small datasets, we show that importance sampling becomes a tractable, unbiased, and single-pass posterior inference method. For settings requiring exploration, we propose a population-based strategy that decomposes the posterior into a sequence of annealed distributions, serving as a new remedy for poor mixing in Energy-based Models. KAEM attains competitive Fr'echet Inception Distance among latent-prior models on SVHN, CIFAR10, and CelebA while sampling in a single forward pass at lower cost than iterative EBMs, and exposing an interpretable prior built from 1D densities.