  The restricted Boltzmann machine is a basic machine learning tool able in
principle to model the distribution of some arbitrary dataset.
  Its standard training procedure appears however delicate and obscure in many
respects.
  We bring some new insights on it by considering the situation where the data
have low intrinsic dimension, offering the possibility of an exact treatment,
illustrated with simple study cases where the standard training is observed to
fail. The reason for this failure is clarified thanks to a Coulomb interactions
reformulation of the model where the hidden features are uniformly charged
hyperplanes interacting repulsively with point-like charges representing the
data. This leads us to consider a convex relaxation of the original
optimization problem thereby resulting in a unique solution, obtained in
precise numerical form in $d=1,2$ while a constrained linear regression
solution is conjectured on the basis of an information theory argument.
