  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.
