  We study the high-dimensional limit of the free energy associated with the
inference problem of finite-rank matrix tensor products. In general, we bound
the limit from above by the unique solution to a certain Hamilton-Jacobi
equation. Under additional assumptions on the nonlinearity in the equation
which is determined explicitly by the model, we identify the limit with the
solution. Two notions of solutions, weak solutions and viscosity solutions, are
considered, each of which has its own advantages and requires different
treatments. For concreteness, we apply our results to a model with i.i.d.
entries and symmetric interactions. In particular, for the first order and even
order tensor products, we identify the limit and obtain estimates on
convergence rates; for other odd orders, upper bounds are obtained.
