# GumBolt: Extending Gumbel trick to Boltzmann priors

arXiv: https://arxiv.org/abs/1805.07349 · HTML: https://mlsanity.com/paper/1805.07349/

## Key facts

- Cited by 9 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 18 May 2018.

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

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- [Non-native Quantum Generative Optimization with Adversarial Autoencoders](https://mlsanity.com/paper/2407.13830/) (2024)
- [A Markov Random Field Multi-Modal Variational AutoEncoder](https://mlsanity.com/paper/2408.09576/) (2024)
- [Multi-Component VAE with Gaussian Markov Random Field](https://mlsanity.com/paper/2507.12165/) (2025)
- [Anomaly Detection in Aeronautics Data with Quantum-compatible Discrete Deep Generative Model](https://mlsanity.com/paper/2303.12302/) (2023)
- [Molecular Design beyond Training Data with Novel Extended Objective Functionals of Generative AI Models Driven by Quantum Annealing Computer](https://mlsanity.com/paper/2602.15451/) (2026)

## Abstract (arXiv)

Boltzmann machines (BMs) are appealing candidates for powerful priors in variational autoencoders (VAEs), as they are capable of capturing nontrivial and multi-modal distributions over discrete variables. However, non-differentiability of the discrete units prohibits using the reparameterization trick, essential for low-noise back propagation. The Gumbel trick resolves this problem in a consistent way by relaxing the variables and distributions, but it is incompatible with BM priors. Here, we propose the GumBolt, a model that extends the Gumbel trick to BM priors in VAEs. GumBolt is significantly simpler than the recently proposed methods with BM prior and outperforms them by a considerable margin. It achieves state-of-the-art performance on permutation invariant MNIST and OMNIGLOT datasets in the scope of models with only discrete latent variables. Moreover, the performance can be further improved by allowing multi-sampled (importance-weighted) estimation of log-likelihood in training, which was not possible with previous models.