  We compute the large-scale limit of the free energy associated with the
problem of inference of a finite-rank matrix. The method follows the principle
put forward in arXiv:1811.01432 which consists in identifying a suitable
Hamilton-Jacobi equation satisfied by the limit free energy. We simplify the
approach of arXiv:1811.01432 using a notion of weak solution of the
Hamilton-Jacobi equation which is more convenient to work with and is
applicable whenever the non-linearity in the equation is convex.
