# Deep Energy Estimator Networks

arXiv: https://arxiv.org/abs/1805.08306 · HTML: https://mlsanity.com/paper/1805.08306/

## Key facts

- Cited by 34 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 21 May 2018.

## Built on by (most cited first, 30 of 34)

- [Image Super-Resolution via Iterative Refinement](https://mlsanity.com/paper/2104.07636/) (2021)
- [Score-based Generative Modeling in Latent Space](https://mlsanity.com/paper/2106.05931/) (2021)
- [Understanding Diffusion Models: A Unified Perspective](https://mlsanity.com/paper/2208.11970/) (2022)
- [Deep Generative Modelling: A Comparative Review of VAEs, GANs, Normalizing Flows, Energy-Based and Autoregressive Models](https://mlsanity.com/paper/2103.04922/) (2021)
- [Diffusion Models for Video Prediction and Infilling](https://mlsanity.com/paper/2206.07696/) (2022)
- [SE(3)-DiffusionFields: Learning smooth cost functions for joint grasp and motion optimization through diffusion](https://mlsanity.com/paper/2209.03855/) (2022)
- [A Complete Survey on Generative AI (AIGC): Is ChatGPT from GPT-4 to GPT-5 All You Need?](https://mlsanity.com/paper/2303.11717/) (2023)
- [Estimating High Order Gradients of the Data Distribution by Denoising](https://mlsanity.com/paper/2111.04726/) (2021)
- [Optimal score estimation via empirical Bayes smoothing](https://mlsanity.com/paper/2402.07747/) (2024)
- [Diffusion-Based Representation Learning](https://mlsanity.com/paper/2105.14257/) (2021)
- [Energy-Based Imitation Learning](https://mlsanity.com/paper/2004.09395/) (2020)
- [Protein Discovery with Discrete Walk-Jump Sampling](https://mlsanity.com/paper/2306.12360/) (2023)
- [Probabilistic Adaptation of Text-to-Video Models](https://mlsanity.com/paper/2306.01872/) (2023)
- [Arbitrary Conditional Distributions with Energy](https://mlsanity.com/paper/2102.04426/) (2021)
- [Semi-Implicit Variational Inference via Score Matching](https://mlsanity.com/paper/2308.10014/) (2023)
- [Learning normalized image densities via dual score matching](https://mlsanity.com/paper/2506.05310/) (2025)
- [On Investigating the Conservative Property of Score-Based Generative Models](https://mlsanity.com/paper/2209.12753/) (2022)
- [Aligning Diffusion Behaviors with Q-functions for Efficient Continuous Control](https://mlsanity.com/paper/2407.09024/) (2024)
- [From Points to Functions: Infinite-dimensional Representations in Diffusion Models](https://mlsanity.com/paper/2210.13774/) (2022)
- [Building Minimal and Reusable Causal State Abstractions for Reinforcement Learning](https://mlsanity.com/paper/2401.12497/) (2024)
- [Multimeasurement Generative Models](https://mlsanity.com/paper/2112.09822/) (2021)
- [Anomaly Detection and Generation with Diffusion Models: A Survey](https://mlsanity.com/paper/2506.09368/) (2025)
- [PnP-ReG: Learned Regularizing Gradient for Plug-and-Play Gradient Descent](https://mlsanity.com/paper/2204.13940/) (2022)
- [Moment Matching Denoising Gibbs Sampling](https://mlsanity.com/paper/2305.11650/) (2023)
- [Universal Smoothed Score Functions for Generative Modeling](https://mlsanity.com/paper/2303.11669/) (2023)
- [Persistently Trained, Diffusion-assisted Energy-based Models](https://mlsanity.com/paper/2304.10707/) (2023)
- [ProtFlow: Fast Protein Sequence Design via Flow Matching on Compressed Protein Language Model Embeddings](https://mlsanity.com/paper/2504.10983/) (2025)
- [Knowledge Transfer in Deep Reinforcement Learning for Slice-Aware Mobility Robustness Optimization](https://mlsanity.com/paper/2203.03227/) (2022)
- [Automatic design of novel potential 3CLpro and PLpro inhibitors](https://mlsanity.com/paper/2101.11890/) (2021)
- [Diffusion Based Causal Representation Learning](https://mlsanity.com/paper/2311.05421/) (2023)

## Abstract (arXiv)

Density estimation is a fundamental problem in statistical learning. This problem is especially challenging for complex high-dimensional data due to the curse of dimensionality. A promising solution to this problem is given here in an inference-free hierarchical framework that is built on score matching. We revisit the Bayesian interpretation of the score function and the Parzen score matching, and construct a multilayer perceptron with a scalable objective for learning the energy (i.e. the unnormalized log-density), which is then optimized with stochastic gradient descent. In addition, the resulting deep energy estimator network (DEEN) is designed as products of experts. We present the utility of DEEN in learning the energy, the score function, and in single-step denoising experiments for synthetic and high-dimensional data. We also diagnose stability problems in the direct estimation of the score function that had been observed for denoising autoencoders.