# Semi-Autoregressive Energy Flows: Exploring Likelihood-Free Training of Normalizing Flows

arXiv: https://arxiv.org/abs/2206.06672 · HTML: https://mlsanity.com/paper/2206.06672/

## Key facts

- Cited by 9 papers in the map (data through 14 Oct 2026).
- Builds on 29 of its 54 references that are in the map.
- First posted on arXiv 14 Jun 2022.

## Builds on (most cited first, 29 of 29)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Deep Learning Face Attributes in the Wild](https://mlsanity.com/paper/1411.7766/) (2014)
- [Variational Inference with Normalizing Flows](https://mlsanity.com/paper/1505.05770/) (2015)
- [Density estimation using Real NVP](https://mlsanity.com/paper/1605.08803/) (2016)
- [WaveNet: A Generative Model for Raw Audio](https://mlsanity.com/paper/1609.03499/) (2016)
- [Glow: Generative Flow with Invertible 1x1 Convolutions](https://mlsanity.com/paper/1807.03039/) (2018)
- [Normalizing Flows for Probabilistic Modeling and Inference](https://mlsanity.com/paper/1912.02762/) (2019)
- [Fast and Accurate Deep Network Learning by Exponential Linear Units (ELUs)](https://mlsanity.com/paper/1511.07289/) (2015)
- [Conditional Image Generation with PixelCNN Decoders](https://mlsanity.com/paper/1606.05328/) (2016)
- [Masked Autoregressive Flow for Density Estimation](https://mlsanity.com/paper/1705.07057/) (2017)
- [Improving Variational Inference with Inverse Autoregressive Flow](https://mlsanity.com/paper/1606.04934/) (2016)
- [Neural Spline Flows](https://mlsanity.com/paper/1906.04032/) (2019)
- [FFJORD: Free-form Continuous Dynamics for Scalable Reversible Generative Models](https://mlsanity.com/paper/1810.01367/) (2018)
- [Sliced Score Matching: A Scalable Approach to Density and Score Estimation](https://mlsanity.com/paper/1905.07088/) (2019)
- [Invertible Residual Networks](https://mlsanity.com/paper/1811.00995/) (2018)
- [Generative Moment Matching Networks](https://mlsanity.com/paper/1502.02761/) (2015)
- [Neural Autoregressive Flows](https://mlsanity.com/paper/1804.00779/) (2018)
- [Generalized Sliced Wasserstein Distances](https://mlsanity.com/paper/1902.00434/) (2019)
- [Training generative neural networks via Maximum Mean Discrepancy optimization](https://mlsanity.com/paper/1505.03906/) (2015)
- [The Cramer Distance as a Solution to Biased Wasserstein Gradients](https://mlsanity.com/paper/1705.10743/) (2017)
- [Unconstrained Monotonic Neural Networks](https://mlsanity.com/paper/1908.05164/) (2019)
- [SurVAE Flows: Surjections to Bridge the Gap between VAEs and Flows](https://mlsanity.com/paper/2007.02731/) (2020)
- [Distributional Sliced-Wasserstein and Applications to Generative Modeling](https://mlsanity.com/paper/2002.07367/) (2020)
- [Rectangular Flows for Manifold Learning](https://mlsanity.com/paper/2106.01413/) (2021)
- [A Tale of Two Flows: Cooperative Learning of Langevin Flow and Normalizing Flow Toward Energy-Based Model](https://mlsanity.com/paper/2205.06924/) (2022)
- [Hybrid Models with Deep and Invertible Features](https://mlsanity.com/paper/1902.02767/) (2019)
- [Invertible Generative Modeling using Linear Rational Splines](https://mlsanity.com/paper/2001.05168/) (2020)
- [Autoregressive Quantile Flows for Predictive Uncertainty Estimation](https://mlsanity.com/paper/2112.04643/) (2021)
- [Self Normalizing Flows](https://mlsanity.com/paper/2011.07248/) (2020)

## Built on by (most cited first, 9 of 9)

- [Simple and Effective Masked Diffusion Language Models](https://mlsanity.com/paper/2406.07524/) (2024)
- [Block Diffusion: Interpolating Between Autoregressive and Diffusion Language Models](https://mlsanity.com/paper/2503.09573/) (2025)
- [Diffusion Models With Learned Adaptive Noise](https://mlsanity.com/paper/2312.13236/) (2023)
- [DySLIM: Dynamics Stable Learning by Invariant Measure for Chaotic Systems](https://mlsanity.com/paper/2402.04467/) (2024)
- [Distributional Diffusion Models with Scoring Rules](https://mlsanity.com/paper/2502.02483/) (2025)
- [Quasi-Bayes meets Vines](https://mlsanity.com/paper/2406.12764/) (2024)
- [Denoising Diffusion Variational Inference: Diffusion Models as Expressive Variational Posteriors](https://mlsanity.com/paper/2401.02739/) (2024)
- [On Measuring Intrinsic Causal Attributions in Deep Neural Networks](https://mlsanity.com/paper/2505.09660/) (2025)
- [Investigating Calibration Challenges in Probabilistic Electricity Price Forecasting](https://mlsanity.com/paper/2606.09517/) (2026)

## Abstract (arXiv)

Training normalizing flow generative models can be challenging due to the need to calculate computationally expensive determinants of Jacobians. This paper studies the likelihood-free training of flows and proposes the energy objective, an alternative sample-based loss based on proper scoring rules. The energy objective is determinant-free and supports flexible model architectures that are not easily compatible with maximum likelihood training, including semi-autoregressive energy flows, a novel model family that interpolates between fully autoregressive and non-autoregressive models. Energy flows feature competitive sample quality, posterior inference, and generation speed relative to likelihood-based flows; this performance is decorrelated from the quality of log-likelihood estimates, which are generally very poor. Our findings question the use of maximum likelihood as an objective or a metric, and contribute to a scientific study of its role in generative modeling.