# Wasserstein Training of Boltzmann Machines

arXiv: https://arxiv.org/abs/1507.01972 · HTML: https://mlsanity.com/paper/1507.01972/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 7 Jul 2015.

## Built on by (most cited first, 3 of 3)

- [Information Geometry of Wasserstein Statistics on Shapes and Affine Deformations](https://mlsanity.com/paper/2307.12508/) (2023)
- [Efficient estimates of optimal transport via low-dimensional embeddings](https://mlsanity.com/paper/2111.04838/) (2021)
- [Fast L2 optimal mass transport via reduced basis methods for the Monge-Ampgraverm ere equation](https://mlsanity.com/paper/2112.01878/) (2021)

## Abstract (arXiv)

The Boltzmann machine provides a useful framework to learn highly complex, multimodal and multiscale data distributions that occur in the real world. The default method to learn its parameters consists of minimizing the Kullback-Leibler (KL) divergence from training samples to the Boltzmann model. We propose in this work a novel approach for Boltzmann training which assumes that a meaningful metric between observations is given. This metric can be represented by the Wasserstein distance between distributions, for which we derive a gradient with respect to the model parameters. Minimization of this new Wasserstein objective leads to generative models that are better when considering the metric and that have a cluster-like structure. We demonstrate the practical potential of these models for data completion and denoising, for which the metric between observations plays a crucial role.