PH-VAE - ICML 2026

Paper:

• arXiv version

Phase-Type Variational Autoencoder for Heavy-Tailed Data
A PyTorch implementation of a multivariate Phase-Type Variational Autoencoder (PH-VAE) for modeling positive-valued continuous and heavy-tailed data.

This work bridges:

• Deep generative modeling
• Applied probability
• Phase-Type distributions
• Heavy-tail statistical modeling

and explores their integration within modern latent-variable learning frameworks.

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PH-VAE architecture with latent-conditioned Phase-Type decoder.

PH-VAE replaces the classical Gaussian decoder of a Variational Autoencoder with a latent-conditioned Phase-Type (PH) distribution. Instead of assuming a fixed parametric likelihood, the decoder learns flexible stochastic processes capable of modeling:

• Heavy-tailed behavior
• Skewed distributions
• Extreme quantiles
• Multivariate dependence through shared latent variables

The repository contains:

• A complete PyTorch implementation of PH-VAE
• Phase-Type likelihood evaluation utilities
• ELBO-based training
• PH sampling routines
• Exploratory notebooks on synthetic and real-world datasets

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What the Repo contains:

• Multivariate PH-VAE implementation in PyTorch
• Latent-conditioned Phase-Type decoder
• Exact PH likelihood computation
• Matrix exponential + uniformization methods
• ELBO optimization with KL regularization
• Sampling utilities for Phase-Type distributions
• Heavy-tail modeling for positive-valued data

Installation

1. Create a Python Environment

Python 3.10+ is recommended.

python -m venv ph_env
source ph_env/bin/activate

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2. Install Dependencies

Install the project dependencies from the repository root:

pip install -r requirements.txt

If you plan to run the notebooks, you may also want:

pip install jupyter ipykernel

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Usage

Running the Project

Open the repository in:

• Jupyter Notebook
• VS Code
• JupyterLab

and run one of the notebooks inside notebooks/.

Important: run notebooks from the repository root so imports resolve correctly.

Example imports:

from models import MultiDimPHVAE
from utils import *

If running from inside notebooks/, change the working directory to the repository root first. First cell does that...

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Example Training

from models import MultiDimPHVAE
from models.ph_vae.ph_vae_trainer import train_phvae

model = MultiDimPHVAE(
    input_dim=dim,
    latent_dim=10,
    n_phases=15
)

train_phvae(
    model,
    train_loader,
    epochs=100,
    learning_rate=1e-3
)

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Checkpoints

A pretrained checkpoint with Weibull data is available in:

models/saved/phvae_weibull.pt

It can be used for:

• Inference
• Sampling
• Evaluation
• Visualization

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Example Applications

• Financial risk modeling
• Insurance losses
• Reliability analysis
• Word-frequency modeling
• Rare-event generative modeling

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Citation

@article{ziani2026phvae,
  title={Phase-Type Variational Autoencoders for Heavy-Tailed Data},
  author={Ziani, Abdelhakim and Horvath, Andras and Ballarini, Paolo},
  journal={arXiv preprint arXiv:2603.01800},
  year={2026}
}

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