# On the Generative Utility of Cyclic Conditionals

arXiv: https://arxiv.org/abs/2106.15962 · HTML: https://mlsanity.com/paper/2106.15962/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 54 of its 98 references that are in the map.
- First posted on arXiv 30 Jun 2021.

## Builds on (most cited first, 30 of 54)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [Deep Unsupervised Learning using Nonequilibrium Thermodynamics](https://mlsanity.com/paper/1503.03585/) (2015)
- [Generative Modeling by Estimating Gradients of the Data Distribution](https://mlsanity.com/paper/1907.05600/) (2019)
- [Auto-Encoding Variational Bayes](https://mlsanity.com/paper/1312.6114/) (2013)
- [Variational Inference with Normalizing Flows](https://mlsanity.com/paper/1505.05770/) (2015)
- [Glow: Generative Flow with Invertible 1x1 Convolutions](https://mlsanity.com/paper/1807.03039/) (2018)
- [DiffWave: A Versatile Diffusion Model for Audio Synthesis](https://mlsanity.com/paper/2009.09761/) (2020)
- [NICE: Non-linear Independent Components Estimation](https://mlsanity.com/paper/1410.8516/) (2014)
- [FastSpeech 2: Fast and High-Quality End-to-End Text to Speech](https://mlsanity.com/paper/2006.04558/) (2020)
- [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)
- [NVAE: A Deep Hierarchical Variational Autoencoder](https://mlsanity.com/paper/2007.03898/) (2020)
- [Generating Sentences from a Continuous Space](https://mlsanity.com/paper/1511.06349/) (2015)
- [FFJORD: Free-form Continuous Dynamics for Scalable Reversible Generative Models](https://mlsanity.com/paper/1810.01367/) (2018)
- [Stein Variational Gradient Descent: A General Purpose Bayesian Inference Algorithm](https://mlsanity.com/paper/1608.04471/) (2016)
- [Invertible Residual Networks](https://mlsanity.com/paper/1811.00995/) (2018)
- [Adversarial Feature Learning](https://mlsanity.com/paper/1605.09782/) (2016)
- [Stochastic Gradient Hamiltonian Monte Carlo](https://mlsanity.com/paper/1402.4102/) (2014)
- [DualGAN: Unsupervised Dual Learning for Image-to-Image Translation](https://mlsanity.com/paper/1704.02510/) (2017)
- [Learning to Discover Cross-Domain Relations with Generative Adversarial Networks](https://mlsanity.com/paper/1703.05192/) (2017)
- [Residual Flows for Invertible Generative Modeling](https://mlsanity.com/paper/1906.02735/) (2019)
- [Causal Discovery with Continuous Additive Noise Models](https://mlsanity.com/paper/1309.6779/) (2013)
- [Analyzing Inverse Problems with Invertible Neural Networks](https://mlsanity.com/paper/1808.04730/) (2018)
- [Adversarially Learned Inference](https://mlsanity.com/paper/1606.00704/) (2016)
- [Diagnosing and Enhancing VAE Models](https://mlsanity.com/paper/1903.05789/) (2019)
- [Hyperspherical Variational Auto-Encoders](https://mlsanity.com/paper/1804.00891/) (2018)
- [CausalGAN: Learning Causal Implicit Generative Models with Adversarial Training](https://mlsanity.com/paper/1709.02023/) (2017)

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

- [Invertible Rescaling Network and Its Extensions](https://mlsanity.com/paper/2210.04188/) (2022)
- [Listening to the Noise: Blind Denoising with Gibbs Diffusion](https://mlsanity.com/paper/2402.19455/) (2024)
- [Symmetric Equilibrium Learning of VAEs](https://mlsanity.com/paper/2307.09883/) (2023)
- [Deep Multivariate Models with Parametric Conditionals](https://mlsanity.com/paper/2602.01953/) (2026)

## Abstract (arXiv)

We study whether and how can we model a joint distribution p(x,z) using two conditional models p(x|z) and q(z|x) that form a cycle. This is motivated by the observation that deep generative models, in addition to a likelihood model p(x|z), often also use an inference model q(z|x) for extracting representation, but they rely on a usually uninformative prior distribution p(z) to define a joint distribution, which may render problems like posterior collapse and manifold mismatch. To explore the possibility to model a joint distribution using only p(x|z) and q(z|x), we study their compatibility and determinacy, corresponding to the existence and uniqueness of a joint distribution whose conditional distributions coincide with them. We develop a general theory for operable equivalence criteria for compatibility, and sufficient conditions for determinacy. Based on the theory, we propose a novel generative modeling framework CyGen that only uses the two cyclic conditional models. We develop methods to achieve compatibility and determinacy, and to use the conditional models to fit and generate data. With the prior constraint removed, CyGen better fits data and captures more representative features, supported by both synthetic and real-world experiments.