  We study the high-dimensional limit of the free energy associated with the
inference problem of a rank-one nonsymmetric matrix. The matrix is expressed as
the outer product of two vectors, not necessarily independent. The
distributions of the two vectors are only assumed to have scaled bounded
supports. We bound the difference between the free energy and the solution to a
suitable Hamilton-Jacobi equation in terms of two much simpler quantities:
concentration rate of this free energy, and the convergence rate of a simpler
free energy in a decoupled system. To demonstrate the versatility of this
approach, we apply our result to the i.i.d. case and the spherical case. By
plugging in estimates of the two simpler quantities, we identify the limits and
obtain convergence rates.
