# Variational Autoencoder with Implicit Optimal Priors

arXiv: https://arxiv.org/abs/1809.05284 · HTML: https://mlsanity.com/paper/1809.05284/

## Key facts

- Cited by 33 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 14 Sep 2018.

## Built on by (most cited first, 30 of 33)

- [Score-based Generative Modeling in Latent Space](https://mlsanity.com/paper/2106.05931/) (2021)
- [LION: Latent Point Diffusion Models for 3D Shape Generation](https://mlsanity.com/paper/2210.06978/) (2022)
- [D2C: Diffusion-Denoising Models for Few-shot Conditional Generation](https://mlsanity.com/paper/2106.06819/) (2021)
- [EQ-VAE: Equivariance Regularized Latent Space for Improved Generative Image Modeling](https://mlsanity.com/paper/2502.09509/) (2025)
- [Contrastively Disentangled Sequential Variational Autoencoder](https://mlsanity.com/paper/2110.12091/) (2021)
- [Factor Analysis, Probabilistic Principal Component Analysis, Variational Inference, and Variational Autoencoder: Tutorial and Survey](https://mlsanity.com/paper/2101.00734/) (2021)
- [Variational Multi-Task Learning with Gumbel-Softmax Priors](https://mlsanity.com/paper/2111.05323/) (2021)
- [Continual Variational Autoencoder Learning via Online Cooperative Memorization](https://mlsanity.com/paper/2207.10131/) (2022)
- [A prior-based approximate latent Riemannian metric](https://mlsanity.com/paper/2103.05290/) (2021)
- [Diffusion Models with Deterministic Normalizing Flow Priors](https://mlsanity.com/paper/2309.01274/) (2023)
- [Meta-Learning for Relative Density-Ratio Estimation](https://mlsanity.com/paper/2107.00801/) (2021)
- [Latent Space Diffusion Models of Cryo-EM Structures](https://mlsanity.com/paper/2211.14169/) (2022)
- [Coupled Variational Autoencoder](https://mlsanity.com/paper/2306.02565/) (2023)
- [Learning Latent Space Hierarchical EBM Diffusion Models](https://mlsanity.com/paper/2405.13910/) (2024)
- [Diffusion Bridge AutoEncoders for Unsupervised Representation Learning](https://mlsanity.com/paper/2405.17111/) (2024)
- [Learning Graphon Autoencoders for Generative Graph Modeling](https://mlsanity.com/paper/2105.14244/) (2021)
- [Autoencoders as Tools for Program Synthesis](https://mlsanity.com/paper/2108.07129/) (2021)
- [Disentangling Autoencoders (DAE)](https://mlsanity.com/paper/2202.09926/) (2022)
- [Variational autoencoder with weighted samples for high-dimensional non-parametric adaptive importance sampling](https://mlsanity.com/paper/2310.09194/) (2023)
- [Nonparametric Automatic Differentiation Variational Inference with Spline Approximation](https://mlsanity.com/paper/2403.06302/) (2024)
- [JADE: Joint-aware Latent Diffusion for 3D Human Generative Modeling](https://mlsanity.com/paper/2412.20470/) (2024)
- [VIVAT: Virtuous Improving VAE Training through Artifact Mitigation](https://mlsanity.com/paper/2506.07863/) (2025)
- [Neural Variational Dropout Processes](https://mlsanity.com/paper/2510.19425/) (2025)
- [Deep clustering with fusion autoencoder](https://mlsanity.com/paper/2201.04727/) (2022)
- [Sparse Representation Learning with Modified q-VAE towards Minimal Realization of World Model](https://mlsanity.com/paper/2208.03936/) (2022)
- [Variational Graph Auto-Encoder Based Inductive Learning Method for Semi-Supervised Classification](https://mlsanity.com/paper/2403.17500/) (2024)
- [Flexible Bayesian Last Layer Models Using Implicit Priors and Diffusion Posterior Sampling](https://mlsanity.com/paper/2408.03746/) (2024)
- [Heavy-Tailed Class-Conditional Priors for Long-Tailed Generative Modeling](https://mlsanity.com/paper/2509.02154/) (2025)
- [Semantic-Aware Prefix Learning for Token-Efficient Image Generation](https://mlsanity.com/paper/2603.25249/) (2026)
- [DOI: Divergence-based Out-of-Distribution Indicators via Deep Generative Models](https://mlsanity.com/paper/2108.05509/) (2021)

## Abstract (arXiv)

The variational autoencoder (VAE) is a powerful generative model that can estimate the probability of a data point by using latent variables. In the VAE, the posterior of the latent variable given the data point is regularized by the prior of the latent variable using Kullback Leibler (KL) divergence. Although the standard Gaussian distribution is usually used for the prior, this simple prior incurs over-regularization. As a sophisticated prior, the aggregated posterior has been introduced, which is the expectation of the posterior over the data distribution. This prior is optimal for the VAE in terms of maximizing the training objective function. However, KL divergence with the aggregated posterior cannot be calculated in a closed form, which prevents us from using this optimal prior. With the proposed method, we introduce the density ratio trick to estimate this KL divergence without modeling the aggregated posterior explicitly. Since the density ratio trick does not work well in high dimensions, we rewrite this KL divergence that contains the high-dimensional density ratio into the sum of the analytically calculable term and the low-dimensional density ratio term, to which the density ratio trick is applied. Experiments on various datasets show that the VAE with this implicit optimal prior achieves high density estimation performance.