# Phase-Type Variational Autoencoders for Heavy-Tailed Data

arXiv: https://arxiv.org/abs/2603.01800 · HTML: https://mlsanity.com/paper/2603.01800/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 16 of its 54 references that are in the map.
- First posted on arXiv 2 Mar 2026.

## Builds on (most cited first, 16 of 16)

- [An Introduction to Variational Autoencoders](https://mlsanity.com/paper/1906.02691/) (2019)
- [Tutorial on Variational Autoencoders](https://mlsanity.com/paper/1606.05908/) (2016)
- [VAE with a VampPrior](https://mlsanity.com/paper/1705.07120/) (2017)
- [Variational Lossy Autoencoder](https://mlsanity.com/paper/1611.02731/) (2016)
- [Learning Hierarchical Priors in VAEs](https://mlsanity.com/paper/1905.04982/) (2019)
- [t3-Variational Autoencoder: Learning Heavy-tailed Data with Student's t and Power Divergence](https://mlsanity.com/paper/2312.01133/) (2023)
- [Variational Mixture of Normalizing Flows](https://mlsanity.com/paper/2009.00585/) (2020)
- [A VAE Approach to Sample Multivariate Extremes](https://mlsanity.com/paper/2306.10987/) (2023)
- [Generation and Simulation of Synthetic Datasets with Copulas](https://mlsanity.com/paper/2203.17250/) (2022)
- [Hierarchical VAE with a Diffusion-based VampPrior](https://mlsanity.com/paper/2412.01373/) (2024)
- [Internet Traffic Volumes Are Not Gaussian -- They Are Log-Normal: An 18-Year Longitudinal Study With Implications for Modelling and Prediction (Complete Version)](https://mlsanity.com/paper/2007.10150/) (2020)
- [Zipf Distributions from Two-Stage Symbolic Processes: Stability Under Stochastic Lexical Filtering](https://mlsanity.com/paper/2511.21060/) (2025)
- [Heavy-Tailed Distribution of the Number of Papers within Scientific Journals](https://mlsanity.com/paper/2011.05703/) (2020)
- [From Bits to Insights: Exploring Network Traffic, Traffic Matrices, and Heavy-Tailed Data](https://mlsanity.com/paper/2311.03562/) (2023)
- [Capturing Extreme Events in Turbulence using an Extreme Variational Autoencoder (xVAE)](https://mlsanity.com/paper/2502.04685/) (2025)
- [Heavy-Tailed Class-Conditional Priors for Long-Tailed Generative Modeling](https://mlsanity.com/paper/2509.02154/) (2025)

## Abstract (arXiv)

Heavy-tailed distributions are ubiquitous in real-world data, where rare but extreme events dominate risk and variability. However, standard Variational Autoencoders (VAEs) employ simple decoder distributions, such as Gaussian distributions, that fail to capture heavy-tailed behavior, while existing heavy-tail-aware extensions remain restricted to predefined parametric families whose tail behavior is fixed a priori. We propose the Phase-Type Variational Autoencoder (PH-VAE), whose decoder distribution is a latent-conditioned Phase-Type (PH) distribution, defined as the absorption time of a continuous-time Markov chain (CTMC). This formulation composes multiple exponential time scales, yielding a flexible and analytically tractable decoder that adapts its finite-range tail behavior directly from the observed data. Experiments on synthetic and real-world benchmarks demonstrate that PH-VAE accurately approximates diverse heavy-tailed distributions, significantly outperforming Gaussian, Student-t, and extreme-value-based VAE decoders in modeling observed tail behavior and extreme quantiles. In multivariate settings, PH-VAE captures realistic cross-dimensional tail dependence through its shared latent representation. To our knowledge, this is the first work to integrate Phase-Type distributions into deep generative modeling, bridging applied probability and representation learning.