  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 to it by considering the situation where the data have
low intrinsic dimension, offering the possibility of an exact treatment and
revealing a fundamental failure of the standard training procedure. The reasons
for this failure \textemdash~like the occurrence of first-order phase
transitions during training~\textemdash \ are clarified thanks to a Coulomb
interactions reformulation of the model. In addition a convex relaxation of the
original optimization problem is formulated thereby resulting in a unique
solution, obtained in precise numerical form on $d=1,2$ study cases, while a
constrained linear regression solution can be conjectured on the basis of an
information theory argument.
